Okay, judging by comments here, maybe I'd be better off sticking with Mint. I've only been working with computers for 25 years, but it is difficult to wrap my head around this, especially when it SHOULD be working.
I will set up my Mint computer a bit later and try again, making sure I have it with me as I'm typing.
Many thanks to all for your assistance, it's really appreciated.
To be honest, this situation has me really annoyed because I feel like I should be able to help... but I have no experience with Mint's file manager because I got into Linux early enough that, by the time Mint got invented, I was happily using stuff much further on the power-user/greybeard end of the "ease of use vs. light on system resources" spectrum.
That said, two things I'd suggest are:
1. if you don't mind installing a second file manager for testing purposes, that's one thing you could try to narrow down where the problem is. If another file manager works, you know it's something in the desktop.
(Not necessarily the file manager, because some desktops spit up the responsibilities. For example, KDE is made of a whole bunch of pieces communicating over D-Bus.)
I use PCManFM and, when I double-click an executable .sh file, it pops up a dialog with "Execute", "Execute in Terminal", "Open", and "Cancel" buttons. (I've attached a screenshot as an example.)
2. As part of the "Is it plugged in?" checking:
a. Copy and paste the exact permissions the file has (run `ls -l` in a terminal in the folder containing it and look at the left-most column) here as part of the "Is it plugged in?" checking.
On mine, that'd be this example line:
-rwxrwxr-x 1 ssokolow ssokolow 51M Apr 29 2016 gog_bit_trip_beat_22.214.171.124.sh
b. Run `mount` in a terminal, find the line with the longest matching prefix for the folder containing the installer, and confirm that it doesn't contain "noexec" in the parentheses at the end.
On mine, that'd be this line:
/dev/sdh1 on /mnt/buffalo_ext type ext4 (rw)
c. If you don't want to install a second file manager, you could instead try running it in the terminal by typing or pasting the full path to it, case-sensitive. If that works, that also indicates that it's something in the desktop.
(NOTE: If you don't want to use the full path, you have to prefix things in the current directory with ./ to tell the system that you're not expecting a command searched up from the PATH. That's another protection against exploits based on social engineering.)
Thanks for all your good advice. I installed PCManFM and after a bit of jiggery pokery, it opened both of my executable files without issue. You solved it for me, so thank you very much.
My question is, can I remove the original file manager (Thunar?) and just have PCManFM as my standalone file manager?