You'll need to set up a bat file to feed DOSbox the correct .conf file (shortcuts, unfortunately, don't work very well because they require the drive to always have the same letter).

The following content should do it, no matter where on the drive you place the bat file (substitute paths and filenames as necessary):
cd "\path\to\game\folder"
"..\relative\path\from\game\to\dosbox.exe" -conf config.conf
exit

Or, make a shortcut and add a "-conf xxx.conf" in its launch options. Or it may not work as you stated above. (didn't read the last part). :p

Depends what the launcher exe is calling. You can always compensate by instructing dosbox.conf to auto-mount a relative path and auto-execute the game so you won't need to type that information every time.

Depends on the game. If it's put vital stuff in the registry, it won't work properly if it doesn't have access to that stuff.

I store all my installers and extras on a USB. It turned out to be a good thing when Gog temporarily shut down. I don't know about the full installed game though.