This is a good thread, Darvond!

I am also trying to play the original Tomb Raider 1 (in Linux) and have similar questions. I am intending to try all of the options you have listed and see which one works best for me. Personally, I think for this sort of game, the controls and how it 'feels' is very important, so I am probably going to lean more towards one of the options that is based on the original engine. However, the newer engine re-implementations look very impressive and I will give them a try.

I tried to run OpenLara yesterday and it seems to have a somewhat disappointing lack of documentation. It's not very clear how to set it up and where the data files are supposed to go, so I wasn't able to get it working with the FMVs. I will persist though and post in here if I manage to work it out.

Re. TR 2 & 3: they seem to have rather poor ratings on the Wine AppDB. But those reviews might be a bit old and I will try them anyway to see if I can get them working.

User keyboard and mouse. Start with the first game.
Once you are done use engine modifiers.
What remakes do you have in mind, there is just one - Anniversary - leave it for last.

Just to circle back to this thread:

I tried several options for TR1 in Linux and I ended up using Tomb1Main. It runs very well in Wine and I understand it's quite faithful to the original engine, since it is based on a decompilation. I am very happy with it - I have now played through the whole of TR1 with no issues and am now playing the Unfinished Business levels (which are really good).

I tried TR2 and that seems to work great out of the box for me in Wine. I tried TR3 briefly, but wasn't able to get it working (yet).