Ok, setting up an FTP daemon on a company server, using NoIP to get into it, and get this:

with www.xxxxxx.ddns.net I get:

Hmm. We’re having trouble finding that site.

We can’t connect to the server at www.xxxx.ddns.net.
If that address is correct, here are three other things you can try:

Try again later.
Check your network connection.
If you are connected but behind a firewall, check that Firefox has permission to access the Web.

With xxxxxx.ddns.net I get:

Unable to connect

Firefox can’t establish a connection to the server at xxxx.ddns.net.

The site could be temporarily unavailable or too busy. Try again in a few moments.
If you are unable to load any pages, check your computer’s network connection.
If your computer or network is protected by a firewall or proxy, make sure that Firefox is permitted to access the Web.

Ping gives me this:

kantuck@zeus ~ $ ping xxxx.ddns.net
PING xxxx.ddns.net (24.245.253.199) 56(84) bytes of data.
From bw2pool1-1.xxxx.net (xx.xxx.168.1) icmp_seq=4 Destination Host Unreachable
From bw2pool1-1.xxxx.net (xx.xxx.168.1) icmp_seq=5 Destination Host Unreachable
^C
--- xxxx.ddns.net ping statistics ---
7 packets transmitted, 0 received, +2 errors, 100% packet loss, time 6080ms

kantuck@zeus ~ $

Traceroute gives me:

kantuck@zeus ~ $ traceroute xxx.ddns.net
traceroute to xxxx.ddns.net (xx.xxx.xxx.199), 30 hops max, 60 byte packets
1 bw2pool1-1.xxx.net (xx.xxx.168.1) 22.850 ms 22.854 ms 22.841 ms
2 * * *
3 * * *
4 * * *
5 * * *
6 * * *
7 * * bw2pool1-1.xxx.net (xx.xxx.168.1) 21.437 ms !H
kantuck@zeus ~ $

Finally FTP:

kantuck@zeus ~ $ ftp xxx.ddns.net
ftp: connect: Connection timed out
ftp>

I'm behind a Netgear GS605, and a Motorola SB5101 "Surfboard Cable modem" All servers are on Linux mint 18

I admit I know almost /nothing about networking/, but also suspect it's not one thing. So anything you can advise will help.

I've read that I may need a new router as some are not configurable. I do need a new wired/wireless router anyway, so can you advise if I need a new one?

Welcome to LinuxQuestions.

Is the noip client running on the router or your server? If running on the server did you setup the noip client correctly and is it actually running? If I remember correctly there is a question that asks if it is running behind a router.

Do you know your public IP address? pinging your noip URL should return the public IP address. If not then either the client is not running or configured correctly or has not updated the noip DNS servers yet.

Once that works then you need to setup the router so it forwards whatever services you are running to the IP address of your server. If your computers do not have a static IP address you can setup an address reservation in the router.

Many might question why you want to setup ftp on a company server...