Above.

I want to buy/download all of the following:

Theme Hospital
Wing Commander 3: Heart of the Tiger
Wing Commander 4: Price of Freedom
Wing Commander Prophecy: Gold Edition

However, I don't know if they're compatible or will work on Ubuntu Linux via Lutris? The only GOG game I have right now that works is Soldier of Fortune: Platinum Edition, great game and all but I keep dying near the end of the first level.

I don't know if my answer will be the one you're looking for, because I use Heroic Games Launcher rather than Lutris, but seeing how close the developers of the two projects are working together, I think the user experience should be quite similar.
Most of the time the games just work out of the box. With the titles you mentioned, it seems you're in this category, although some users reported some hiccups.
If they don't work, I go to appdb.winehq.org and protondb.com to see the experience of other users. Usually it's a missing .dll file from Windows, which can be easily installed via Winetricks (it comes built-in with Heroic). The biggest issue with Linux gaming nowadays is anti-cheat, but it seems your taste in games won't be affected by this.

I dont have those games. Assuming they are the DOS versions, all linux major flavors should have some emulators to run those titles. In that case, you would require unpacking the offline windows installer. Sifting out the files you need. Assume gog staff didnt strip anything out. Then run it through the DOS emulator.

Think the program is called innoextract. I wont give the github link, because I never used it.

Gog staff made a mess of something that should be easy. Such as, just having a normal zip file or any other archived type.

In the case of Theme Hospital, CorsixTH is an option: https://corsixth.com/

They should work in Lutris/LinuxMint (not sure if Ubuntu is the same ) & the installation is fully automated as long as you are connected/logged in to your GOG account (any extra runners such as dosbox or Scummvm will install if needed) & are online for the installation. You can disconnect and log out once installer is fininshed. The problem is the GOG versions use regular dosbox and the cycles are not fixed which means you may have to keep increasing/decreasing cycles. If it is dosbox staging then they might not have an issue but i cannot confirm until i have downloaded & tested them.

WC3 is ok. (Select dosbox version on install not wine). Set cpu cycles to 100000/ms, no stuttering or slowdowns in dosbox staging.

WC5 uses Nglide or DGVoodoo but have not got running yet. (could be graphics /resolution issue).

WC4 has no video playback.(DVD version) should be ok with old DOS version.

WC4 Dos version really glitchy kept locking up. Uses regular Dosbox.

Theme Hospital ok in dosbox versions but are other options available.

Hi,

I am also using Ubuntu as my main gaming box, and I can count on one hand the number of GoG games (and non-GoG, for that matter) that do not run perfectly using Bottles (basically, a combined front end that supports Wine, Lutris, Proton, etc). I do not have a huge library, but let's just say that I have had problems getting 1-2% of my library to run. Everything else runs flawlessly. I would be very surprised there is no configuration that would not run the titles you listed.

Good Luck!

Aren't those games all DOS games? They should work just fine with Dosbox then.

Oh, but while you're at it, consider using a version of Dosbox that's a bit fresher than 9 or so years since the last update.

Here we go.