After reading on WineHQ that Tomb Raider runs on Wine now, I did try the demo on my harddisk again. And guess what? It really worked. I played the whole demo, and decided to buy it.
Tomb Raider was the first game I ever bought... It was great that decade ago, and now I felt I wanted to relive the experiece again :)
Installation was a bit tricky: Since I install my games below /games/, and only the root directory was available in wine as drive, setup told me I didn't have 7GiB for installation. Solution was to create another drive that mounts /games/, and installation worked fine. And I had to cd to /mnt/dvd/, otherwise setup wouldn't run.
Running the game did also not work out of the box. I fetched the latest patch and a no-cd crack, and with this (thanks, Deviance!) I could finally start the game. The videos at the beginning take just half of the screen, but the game itself does work really nice.
I had to disable water effects and reflections, since they seem to have some bugs in Wine's D3D implementation, and Lara's shadow is flipped when viewed from the left, but otherwise I'm enjoying a nice action-adventure on my Gentoo box.