When I try to run the Install.sh file, either it crashes or it tells me the the package mono could not be found. Then, I tried to just launch the game through the Launch_Barotrauma.sh, and it immediately crashes. It closes too fast for me to read the terminal output.

If you open the file (Launch_Barotrauma) by clicking on it & selecting open with terminal from the pop up prompt the terminal will close on you when the game crashes, etc.

So to keep the terminal open, your going to need to first open the terminal on its own.
Navigate to the game folder in the terminal.
Then type ./Launch_Barotrauma

That way the terminal will stay open after the game crashes & then you can read what the errors are etc. Most likely your missing a dependency.

I had a similar issue where Install.sh wouldnt do anything and gave me some odd error. I got around this by opening the Install.sh file and copying the command (something like "sudo apt-get filename filename filename mono" ) into the terminal (terminal was opened inside the game folder, I am not sure if this matters). However I had to remove the word "mono" at the end of the command, and install that seperatly by following this http://www.mono-project.com/download/#download-lin (I installed both "mono-devel" and "mono-complete" for no particular reason and am not sure which one is the correct one).

My issue wasnt exactly the same as what you describe and I was installing it onto a chromebook (with linux) so this might not do it for you.
Edit: I reread your question and saw you are using Mint, I dont know anything about Mint so sorry if what I said isnt relevant