The problem is simple: You don't have DirectX 10 or DirectX 11. You can't download it from Microsoft yet, so far they only package it with Vista. That means Eidos is basically forcing gamers to upgrade to Vista in order to play Tomb Raider Underworld. You can download DirectX 10 on the web but it won't be the official version and you won't know if you are downloading viruses and spyware too.
If you have XP there are some webbers offering DirectX 10 online, if you can find it then downloading to your computer will allow you to use the Underworld demo.
Tomb Raider runs fine under XP and Vista. I don't think it uses DX10 for Vista at all. There is no such thing as DX10 for XP nor will there ever be. If you say you installed DX10 in XP I wonder what you actually installed. Maybe that's what screwed things up.
This is one of the best gaming engines for the PC I've seen.
If you still have issues, look for any "DirectX" software and remove it. Make sure your drivers are up to date (graphics, mainboard, audio). Update via Microsoft Update until no more important updates exist. The game should install DX9 if needed. If you still have problems then you've screwed something up beyond my ability to help. Reinstall Windows, your drivers, update via Windows update and install TR again.
It runs smoothly, on my mainstream system at 1360x768 and 60FPS at very high settings. I recommend disabling AA since it rarely makes a difference (on the boat it does, but how often are you there?)
Use FRAPS if needed to tweak settings so you are above 30FPS.
Vista 32-bit and Vista 64-bit are now excellent gaming systems. Most complaints are no longer relevent. I've tested both extensively with all the latest games.
My system: ATI HD3870 512MB, AMD X24800+, 4GBDDR3200(400MHz), Vista 64-bit and XP SP3.
Update: I can't find official specs but it appears that TRU automatically chooses either DX10 or DX9 when in Vista. If you dual-boot and have low frame rates in Vista, try running in XP. Again, use FRAPS to see the frame rates. All current DX10 games also have a DX9 option. I've never used DX10 as the performance benefits outweigh the visual benefits. DX10 seems only like a stepping stone to DX11.
Try the Tomb Raider: Underworld Tech Support forum at the official Eidos site. http://forums.eidosgames.com/forumdisplay.php?f=261
There's even a thread about this very problem. http://forums.eidosgames.com/showthread.php?t=81777
The game requirements may be misleading. I have 2GB RAM, but a site that checks your PC against requirements for specific games said my system didn't have enough free RAM to play Underworld. I'll try to find the link again.
Here's the site for checking your system against game requirements. http://www.systemrequirementslab.com/referrer/srtest
I got the link off of Stella's Tomb Raider Underworld page http://www.tombraiders.net/stella/underworld.html
Which also has links for Tech Support. http://www.tombraiders.net/stella/techlinks.html