Wow, hi Stella! was on your website just today in fact! :P. Think I may have had a chat with you in the past on Tomb Raider forums/chatrooms, actually.
But unfortunately, I don't know. We are having a bit of a conversation down on the Tomb Raider 1 PlayStation boards if you feel like joining in. Two people on those boards in fact mentioned they were going to get TR1 for the PS3. Mainly because he had never played the series and as for the other one, he says that everybody is reporting that their original disc-based TR1 does not work in the PS3. If you have the time, you could join in with our conversations but if not, I will let you know what they say. If they don't have the game yet or haven't replied, I will still let you know within the week :).
The codes work just fine. There's no difference between the original PS1 game and the one you get from PSN seeing how they're the same game. It's harder to do with the PSP if you use the default control setup though, since you need to use the analog stick for L2/R2.