ls -la /etc/resolv.conf
lrwxrwxrwx 1 root root 39 Aug 29 18:36 /etc/resolv.conf -> ../run/systemd/resolve/stub-resolv.conf


It could be related to

https://github.com/systemd/systemd/pull/7014

But it's beyond my level of understanding of the system internals.

Ok, so perhaps a good idea to retrace your steps and check output (as per post #2) again. I don't know enough about systemd-resolved to advise further here, but we should check that the DHCP-assignment is working as expected when connecting to the extender.

EXTENDER

iwconfig wlan0
wlan0 No such device

ip a
1: lo: <LOOPBACK,UP,LOWER_UP> mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000
link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00
inet 127.0.0.1/8 scope host lo
valid_lft forever preferred_lft forever
inet6 ::1/128 scope host
valid_lft forever preferred_lft forever
2: wlp2s0: <BROADCAST,MULTICAST,UP,LOWER_UP> mtu 1500 qdisc noqueue state UP group default qlen 1000
link/ether c4:a8:1d:80:0f:43 brd ff:ff:ff:ff:ff:ff
inet6 2001:1970:549c:a600:50d2:2424:759f:e412/64 scope global temporary dynamic
valid_lft 3599sec preferred_lft 3599sec
inet6 2001:1970:549c:a600:2950:4127:4c2c:2af0/64 scope global dynamic mngtmpaddr noprefixroute
valid_lft 3599sec preferred_lft 3599sec
inet6 fe80::138f:6ba4:86dd:ea3c/64 scope link noprefixroute
valid_lft forever preferred_lft forever

ip r

<blank>

sudo grep -i name /etc/resolv.conf
[sudo] password for roger:
nameserver 127.0.0.53


DIRECT

iwconfig wlan0
wlan0 No such device

ip a
1: lo: <LOOPBACK,UP,LOWER_UP> mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000
link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00
inet 127.0.0.1/8 scope host lo
valid_lft forever preferred_lft forever
inet6 ::1/128 scope host
valid_lft forever preferred_lft forever
2: wlp2s0: <BROADCAST,MULTICAST,UP,LOWER_UP> mtu 1500 qdisc noqueue state UP group default qlen 1000
link/ether c4:a8:1d:80:0f:43 brd ff:ff:ff:ff:ff:ff
inet 192.168.0.22/24 brd 192.168.0.255 scope global dynamic noprefixroute wlp2s0
valid_lft 3051sec preferred_lft 3051sec
inet6 2001:1970:549c:a600:50d2:2424:759f:e412/64 scope global temporary dynamic
valid_lft 3600sec preferred_lft 3600sec
inet6 2001:1970:549c:a600:f9be:1184:d955:4ada/64 scope global dynamic mngtmpaddr noprefixroute
valid_lft 3600sec preferred_lft 3600sec
inet6 fe80::7980:1e6e:60cd:7467/64 scope link noprefixroute
valid_lft forever preferred_lft forever

ip r
default via 192.168.0.1 dev wlp2s0 proto dhcp metric 600
169.254.0.0/16 dev wlp2s0 scope link metric 1000
192.168.0.0/24 dev wlp2s0 proto kernel scope link src 192.168.0.22 metric 600

sudo grep -i name /etc/resolv.conf
[sudo] password for roger:
nameserver 127.0.0.53


Network Manager logs

EXTENDER (after reboot)
Sep 09 19:18:27 roger-Mint NetworkManager[828]: <info> [1536535107.3839] dhcp6 (wlp2s0): state changed bound -> done
Sep 09 19:18:28 roger-Mint NetworkManager[828]: <info> [1536535108.0662] device (wlp2s0): state change: ip-config -> ip-check (reason 'none', sys-iface-state: 'managed')
Sep 09 19:18:28 roger-Mint NetworkManager[828]: <info> [1536535108.0669] device (wlp2s0): state change: ip-check -> secondaries (reason 'none', sys-iface-state: 'managed')
Sep 09 19:18:28 roger-Mint NetworkManager[828]: <info> [1536535108.0671] device (wlp2s0): state change: secondaries -> activated (reason 'none', sys-iface-state: 'managed')
Sep 09 19:18:28 roger-Mint NetworkManager[828]: <info> [1536535108.0672] manager: NetworkManager state is now CONNECTED_LOCAL
Sep 09 19:18:28 roger-Mint NetworkManager[828]: <info> [1536535108.0721] manager: NetworkManager state is now CONNECTED_SITE
Sep 09 19:18:28 roger-Mint NetworkManager[828]: <info> [1536535108.0722] policy: set 'Megatone_5.0' (wlp2s0) as default for IPv6 routing and DNS
Sep 09 19:18:28 roger-Mint NetworkManager[828]: <info> [1536535108.0726] device (wlp2s0): Activation: successful, device activated.
Sep 09 19:18:28 roger-Mint dhclient[1655]: DHCPDISCOVER on wlp2s0 to 255.255.255.255 port 67 interval 7 (xid=0x2bef4f54)
Sep 09 19:18:35 roger-Mint dhclient[1655]: DHCPDISCOVER on wlp2s0 to 255.255.255.255 port 67 interval 14 (xid=0x2bef4f54)
Sep 09 19:18:50 roger-Mint dhclient[1655]: DHCPDISCOVER on wlp2s0 to 255.255.255.255 port 67 interval 14 (xid=0x2bef4f54)
Sep 09 19:18:58 roger-Mint NetworkManager[828]: <info> [1536535138.4175] connectivity: (wlp2s0) timed out
Sep 09 19:19:04 roger-Mint dhclient[1655]: DHCPDISCOVER on wlp2s0 to 255.255.255.255 port 67 interval 14 (xid=0x2bef4f54)
Sep 09 19:19:10 roger-Mint NetworkManager[828]: <warn> [1536535150.4167] dhcp4 (wlp2s0): request timed out
Sep 09 19:19:10 roger-Mint NetworkManager[828]: <info> [1536535150.4168] dhcp4 (wlp2s0): state changed unknown -> timeout
Sep 09 19:19:10 roger-Mint NetworkManager[828]: <info> [1536535150.4329] dhcp4 (wlp2s0): canceled DHCP transaction, DHCP client pid 1655
Sep 09 19:19:10 roger-Mint NetworkManager[828]: <info> [1536535150.4330] dhcp4 (wlp2s0): state changed timeout -> done
Sep 09 19:19:18 roger-Mint NetworkManager[828]: <info> [1536535158.4140] connectivity: (wlp2s0) timed out
Sep 09 19:24:18 roger-Mint NetworkManager[828]: <info> [1536535458.4161] connectivity: (wlp2s0) timed out
Sep 09 19:29:18 roger-Mint NetworkManager[828]: <info> [1536535758.4159] connectivity: (wlp2s0) timed out
Sep 09 19:34:18 roger-Mint NetworkManager[828]: <info> [1536536058.4157] connectivity: (wlp2s0) timed out
Sep 09 19:39:18 roger-Mint NetworkManager[828]: <info> [1536536358.4160] connectivity: (wlp2s0) timed out

DIRECT

Sep 09 19:44:18 roger-Mint NetworkManager[828]: <info> [1536536658.4165] connectivity: (wlp2s0) timed out
Sep 09 19:49:18 roger-Mint NetworkManager[828]: <info> [1536536958.4155] connectivity: (wlp2s0) timed out
Sep 09 19:54:18 roger-Mint NetworkManager[828]: <info> [1536537258.4161] connectivity: (wlp2s0) timed out
Sep 09 19:59:18 roger-Mint NetworkManager[828]: <info> [1536537558.4166] connectivity: (wlp2s0) timed out
Sep 09 20:03:00 roger-Mint NetworkManager[828]: <info> [1536537780.3951] device (wlp2s0): state change: activated -> deactivating (reason 'user-requested', sys-iface-state: 'managed')
Sep 09 20:03:00 roger-Mint NetworkManager[828]: <info> [1536537780.3953] manager: NetworkManager state is now DISCONNECTING
Sep 09 20:03:00 roger-Mint NetworkManager[828]: <info> [1536537780.4012] audit: op="device-disconnect" interface="wlp2s0" ifindex=2 pid=1700 uid=1000 result="success"
Sep 09 20:03:00 roger-Mint NetworkManager[828]: <info> [1536537780.4016] device (wlp2s0): state change: deactivating -> disconnected (reason 'user-requested', sys-iface-state: 'managed')
Sep 09 20:03:00 roger-Mint NetworkManager[828]: <info> [1536537780.4021] dhcp6 (wlp2s0): canceled DHCP transaction
Sep 09 20:03:00 roger-Mint NetworkManager[828]: <info> [1536537780.4034] manager: NetworkManager state is now DISCONNECTED
Sep 09 20:03:00 roger-Mint NetworkManager[828]: <warn> [1536537780.4125] sup-iface[0x560a453b1a40,wlp2s0]: connection disconnected (reason -3)
Sep 09 20:03:00 roger-Mint NetworkManager[828]: <info> [1536537780.4169] device (wlp2s0): supplicant interface state: completed -> disconnected
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4004] device (wlp2s0): Activation: starting connection 'CB38DFD79DA3' (b3c553b9-a16a-4352-a885-28b39222bf4f)
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4006] audit: op="connection-activate" uuid="b3c553b9-a16a-4352-a885-28b39222bf4f" name="CB38DFD79DA3" pid=1700 uid=1000 result="success"
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4008] device (wlp2s0): state change: disconnected -> prepare (reason 'none', sys-iface-state: 'managed')
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4009] manager: NetworkManager state is now CONNECTING
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4014] device (wlp2s0): state change: prepare -> config (reason 'none', sys-iface-state: 'managed')
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4016] device (wlp2s0): Activation: (wifi) access point 'CB38DFD79DA3' has security, but secrets are required.
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4016] device (wlp2s0): state change: config -> need-auth (reason 'none', sys-iface-state: 'managed')
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4017] sup-iface[0x560a453b1a40,wlp2s0]: wps: type pbc start...
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4054] device (wlp2s0): state change: need-auth -> prepare (reason 'none', sys-iface-state: 'managed')
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4062] device (wlp2s0): state change: prepare -> config (reason 'none', sys-iface-state: 'managed')
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4067] device (wlp2s0): Activation: (wifi) connection 'CB38DFD79DA3' has security, and secrets exist. No new secrets needed.
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4068] Config: added 'ssid' value 'CB38DFD79DA3'
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4068] Config: added 'scan_ssid' value '1'
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4068] Config: added 'bgscan' value 'simple:30:-80:86400'
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4069] Config: added 'key_mgmt' value 'WPA-PSK'
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4069] Config: added 'auth_alg' value 'OPEN'
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4069] Config: added 'psk' value '<hidden>'
Sep 09 20:03:13 roger-Mint NetworkManager[828]: <info> [1536537793.4271] device (wlp2s0): supplicant interface state: disconnected -> scanning
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.0622] device (wlp2s0): supplicant interface state: scanning -> authenticating
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.0697] device (wlp2s0): supplicant interface state: authenticating -> associating
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.0797] device (wlp2s0): supplicant interface state: associating -> associated
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.0931] device (wlp2s0): supplicant interface state: associated -> 4-way handshake
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.1110] device (wlp2s0): supplicant interface state: 4-way handshake -> completed
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.1110] device (wlp2s0): Activation: (wifi) Stage 2 of 5 (Device Configure) successful. Connected to wireless network 'CB38DFD79DA3'.
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.1112] device (wlp2s0): state change: config -> ip-config (reason 'none', sys-iface-state: 'managed')
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.1115] dhcp4 (wlp2s0): activation: beginning transaction (timeout in 45 seconds)
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.1136] dhcp4 (wlp2s0): dhclient started with pid 3611
Sep 09 20:03:16 roger-Mint dhclient[3611]: DHCPDISCOVER on wlp2s0 to 255.255.255.255 port 67 interval 3 (xid=0x6a3ae006)
Sep 09 20:03:16 roger-Mint dhclient[3611]: DHCPREQUEST of 192.168.0.22 on wlp2s0 to 255.255.255.255 port 67 (xid=0x6e03a6a)
Sep 09 20:03:16 roger-Mint dhclient[3611]: DHCPOFFER of 192.168.0.22 from 192.168.0.1
Sep 09 20:03:16 roger-Mint dhclient[3611]: DHCPACK of 192.168.0.22 from 192.168.0.1
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.2522] dhcp4 (wlp2s0): address 192.168.0.22
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.2522] dhcp4 (wlp2s0): plen 24 (255.255.255.0)
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.2522] dhcp4 (wlp2s0): gateway 192.168.0.1
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.2522] dhcp4 (wlp2s0): lease time 3600
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.2523] dhcp4 (wlp2s0): hostname 'roger-Mint'
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.2523] dhcp4 (wlp2s0): nameserver '24.226.1.93'
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.2523] dhcp4 (wlp2s0): nameserver '24.226.10.193'
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.2523] dhcp4 (wlp2s0): state changed unknown -> bound
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.2544] device (wlp2s0): state change: ip-config -> ip-check (reason 'none', sys-iface-state: 'managed')
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.2561] device (wlp2s0): state change: ip-check -> secondaries (reason 'none', sys-iface-state: 'managed')
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.2567] device (wlp2s0): state change: secondaries -> activated (reason 'none', sys-iface-state: 'managed')
Sep 09 20:03:16 roger-Mint dhclient[3611]: bound to 192.168.0.22 -- renewal in 1661 seconds.
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.2570] manager: NetworkManager state is now CONNECTED_LOCAL
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.2667] manager: NetworkManager state is now CONNECTED_SITE
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.2668] policy: set 'CB38DFD79DA3' (wlp2s0) as default for IPv4 routing and DNS
Sep 09 20:03:16 roger-Mint NetworkManager[828]: <info> [1536537796.2673] device (wlp2s0): Activation: successful, device activated.
Sep 09 20:03:17 roger-Mint NetworkManager[828]: <info> [1536537797.3656] manager: NetworkManager state is now CONNECTED_GLOBAL
Sep 09 20:03:17 roger-Mint NetworkManager[828]: <info> [1536537797.4277] dhcp6 (wlp2s0): activation: beginning transaction (timeout in 45 seconds)
Sep 09 20:03:17 roger-Mint NetworkManager[828]: <info> [1536537797.4297] dhcp6 (wlp2s0): dhclient started with pid 3716
Sep 09 20:03:17 roger-Mint NetworkManager[828]: <info> [1536537797.4309] policy: set 'CB38DFD79DA3' (wlp2s0) as default for IPv6 routing and DNS
Sep 09 20:03:17 roger-Mint dhclient[3716]: XMT: Info-Request on wlp2s0, interval 920ms.
Sep 09 20:03:17 roger-Mint dhclient[3716]: RCV: Reply message on wlp2s0 from fe80::5e8f:e0ff:fe30:1a59.
Sep 09 20:03:17 roger-Mint NetworkManager[828]: <info> [1536537797.9375] dhcp6 (wlp2s0): nameserver '2001:1970:c06e:c0::94'
Sep 09 20:03:17 roger-Mint NetworkManager[828]: <info> [1536537797.9376] dhcp6 (wlp2s0): nameserver '2001:1970:c06e:c0::93'
Sep 09 20:03:17 roger-Mint NetworkManager[828]: <info> [1536537797.9376] dhcp6 (wlp2s0): nameserver '2001:1970:c0c0:6ec0::194'
Sep 09 20:03:17 roger-Mint NetworkManager[828]: <info> [1536537797.9376] dhcp6 (wlp2s0): state changed unknown -> bound
Sep 09 20:03:17 roger-Mint NetworkManager[828]: <info> [1536537797.9404] dhcp6 (wlp2s0): client pid 3716 exited with status 0
Sep 09 20:03:17 roger-Mint NetworkManager[828]: <info> [1536537797.9404] dhcp6 (wlp2s0): state changed bound -> done

Same as before, some websites are accessible , most are not when connected to the Extender.
However, this is true only after a reboot while connected to the EXTENDER. If I change networks from DIRECT to EXTENDER without rebooting, nothing works.

This reads like IPv6 is working perhaps? You clearly have no IPv4 address assigned. I don't know how to help you further unfortunately. I've never experienced anything like it...

Thank you for all your help, I will use the Direct connection as needed while I'll keep researching your suggestions. I will try as well a different Range Extender to narrow down the possible sources of trouble.

Good idea if you are able.