Well I installed OOO 1.33 FULL for the first time, everything went smoothly, and the game runs fine.

However I noticed that picking up items like bones or a torch, there's no "pickup" noise. This does not occur with Ingredients and weapons. I'm wondering if this is an intended feature or not?

I had this happen once with a new install of Win 7 64-bit. I could hear most sound but not all NPCs. My fix was to change the speaker configuration in Win 7. I changed my speakers to stereo as I use headphones and that fixed it. My guess.

It's not a sound problem. This only occurred after installing OOO.

Here, I'll be more clear: There are no sounds that play if I pick up a bone, or a torch, but if I pick up a knife, there's still a sound.

Haven't checked bones but I definitely get pick up sounds from Torches. Make sure you have http://www.gamesas.com/index.php?/topic/1159634-relz-oscuros-oblivion-overhaul-133/, preferably patched to 1.34 Beta 5. Might be caused by something else if using other mods perhaps.

Well this is unusual: Unchecking all mods and starting a new game, the problem still persists.

I still doubt it's hardware related, as all my other games are fine, so what could it be?

As I mentioned earlier, everything worked for me also in other games and Windows, just not with Oblivion and OOO. Checking the sound speakers configuration fixed it. But if you checked that and it is set correctly, then you figure the problem is somewhere else. I recall a thread recently with someone having a similar issue (except maybe they had no sound at all). It should be in the first 2-3 pages.