Do these versions of TFOA and TLC support the "cheat" for skipping the fist fights? Or do we have to fight them the old fashioned way?

I've played both, GOG version.

In Fate of Atlantis you can press 0 to insta-win a fight (you won't score any IQ points, in case you care about them).

I don't think The Last Crusade ever had this kind of cheat. You can avoid most fights if you choose the appropriate dialogue option (you will have to save and reload a lot, though).

That sound pretty much like I remember it from back in the day. Any hints on how to win the fights in The Last Crusade? I imagine that you could technically win all the fights? Does lost health carry over from one fight to the next?

In The Last Crusade lost health indeed carries over to the next fight. There is a medical aid kit that you can use once to restore all health (however, it can't be used at the zeppelin corridors). It's possible to practice fighting at the boxing ring at Barnett College, but I found I wasn't very good myself. I saved a lot and managed to avoid almost all fights.

My memory of the game is a bit hazy. Are fights limited? Can you beat "all" the enemy soldiers (I'm thinking of the castle and the zeppelin, I think those are the only ones with "random" fights)? Or do they respawn indefinitely?

In the castle once you beat them they don't respawn. In the zeppelin I'm not 100% sure, I'd say it's also the case, but my strategy consists on avoiding them as much as possible so I couldn't tell.

These versions have one major difference from the original releases which does help if you're trying to fight all the guards in TLC or complete the fists path in TFOA - with ScummVM you can save and reload during a fight. So if you put in a few good punches without taking a hit, you can make a save just in case things then don't go your way. It makes getting all the IQ points much easier (you can also save and reload during the biplane sequence in TLC)