If you get a flashing "?" folder when you start the computer it's probably one of two problems. Either it's the harddrive that's corrupt or it's just the OS that's been corrupted for some reason. What I would do is to try to install another harddrive and see if you can reinstall the OS.
If you dropped it in the pool it's probably a gonner. Chlorine is a real electronics killer, but you can try to take it apart and get some really high-proof alcohol (94%+) and just go to town on that phone with it, if you're really lucky you may be able to get it running again, but I wouldn't get my hopes up. -EDIT- Here's a generic guide on how to repair liquid damage in phones.
I'd say it's probably a problem with the cable. http://se.eetgroup.com/i/MSPA2830-Apple-LVDS-cable-2010 It's worth a shot! Edit: Since its working fine with an external monitor, there shouldn't be any problem with the chips. So, either LCD or the LVDS-Cable, and it's not the LCD.... Good luck!
HTC Spare parts are worthless, I'm afraid. For example; I've changed the digitizer/LCD of a HTC One X with 5 different spare parts, all were faulty in some way (dead zones on digitizer, flickering LCD, "wierd" digitizer, etc.). Changing back to the original and everything worked fine. It's the spare part, I'm sorry.
Your backpanel might short-ciruit something in the phone. Try with a 2x2cm-ish piece of paper and put it in different places inside the phone before putting the back panel back on. If it doesn't turn off, you might have found your problem (but not the solution, having a piece of paper in your phone is not a good idea, lol). If you find the component that doesn't want to touch the back panel (yes, components have feelings too... well, at least resistance (OH MY GOD, I'm Hillarious)). Try adhesive tape or something more "professional" if/when you find the problem.