You mention Win10 above, but that partition layout doesn't match - nor the absence of /sys/firmware/efi.
If you can re-install everything, I'd just erase the entire disk and do Win10 first, then Linux. Can't be any more specific.

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What I would do with a complete new install would be use gparted and create a new blank gpt partition table.

I would then set the bios to UEFI and not csm/legacy and do a new install of windows which will partition it the way windows prefers.

Once windows is installed, then from within windows use the disk manager and shrink the windows partition to the size you wish windows to use.

Following that you can do an install of linux in the free space and the partitions should be lined up properly with grub able to boot both linux and windows.

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Thank-you so very much! I will proceed to do that.

Thank-you for this information. In order to create a new blank gpt partition table will the computer do that automatically if I use gparted to Create Partition Table or do I need to do something besides that?

Without intending to be flippant,

Try it and learn. We cannot always hold your hand.
If doing a full disk wipe and reinstall then you cannot hurt anything.

I followed your advice as well as that of computersavvy as best I knew how but it did not go very well; I don't know if the Windows 10 USB is too old or what the matter might be. I selected the gpt option, however, when I started installing Windows 10 it would not work with gpt so had to go to msdos. I turned on UEFI started to install Linux Mint, have created a /efi partition but the system aborted installation due to an error creating the /efi partition. I restarted the LM install but it hung up again creating /data. Here is my fdisk -l output with the sda4 once again not starting on a physical sector boundary. Needless to say it has been a bit frustrating. Thanks for your help.

I just posted the results I got from my efforts.

Oh and I failed to mention that now after attempting to install Linux Mint I cannot boot into Windows 10.

It seems that windows installed in MBR mode and then you installed Mint in UEFI mode. That really does not work.

Try again, with recreating the gpt partition, setting the bios to UEFI, then installing windows. If your windows media still refuses to install in UEFI mode then it appears you only have 2 choices.

1. Use the windows install media you have and install in MBR mode and install Mint in MBR mode as well.

2. Download a new windows install iso so it can be installed in UEFI mode and will be compatible with installing Mint in UEFI mode.

https://www.microsoft.com/en-us/soft...d/windows10ISO

Installing one OS in MBR mode and another in UEFI mode always seems to break booting one or the other.

Thank-you, very helpful! Will try downloading a different Windows iso.

Getting kind of desperate to get a Linux system up and running I decided to basically ignore Windows for the time being and install Linux Mint. I decided to do Create Partition table again to gpt after I had changed the BIOS to UEFI and booted from my USB. When I started the install it had me create a password, presumably for UEFI boot. The install seemed to go without a hitch but now when I try to boot it comes up with the message that there is no bootable media in the system. It would appear to me that the UEFI is stopping it because it never even asks me for that password it had me create at the beginning of the installation. Here is the fdisk -l output: