Beeping usually indicates bad RAM, and not waking up from sleep sounds like it could be caused by that as well. In order to access the RAM you need to take off the screen and access the logic board. Booting to Apple Diagnostics might be able to help diagnose the issue. If you still have AppleCare you might want to have Apple check it out, as opening it will void any warranty it does have.
If your screen was bulging out, it seems more likely due to the battery swelling than the screen being faulty. Have you inspected the battery at all? It could be that it died and won’t take a charge. Also try putting the old screen back on and see if it works, and report back.
You could try PRAM and SMC resets. I doubt those would do much, but it’s worth a shot. Since your computer is only a few months old, you should still be within Apple’s 3 month warranty period. I would take it to an Apple store, they will probably replace it free of charge if they can’t fix it. Taking it apart now would ruin any kind of warranty you would have. To me it looks like a loose display connector, or a chip is coming off the board.
That happened on one of mine as well. It is sometimes possible to take a large flathead screwdriver and turn the mechanism with that rectangular hole in the middle, although it can be a little tricky to do. I ended up resorting to tape because it was too much of a hassle. From what I can tell, there isn’t really a way of fixing that unless you replace the entire bottom case, which is a pretty big task and honestly isn’t worth doing for a computer that old, unless it is really important to you. Here is the guide for that: MacBook Core 2 Duo Lower Case Replacement
If you stick the hard drive in another computer it will boot and you will be able to access your files, although you might have a couple driver issues. This can be fixed by running windows update. Windows will also deactivate, but you can reactivate it using the license key on the bottom of the computer, or it will automatically activate via a product key in the BIOS, depending on the computer. Sounds like your computer is pretty toast though, you could try repairing it, but you are probably better off with a new one at this point.
I have both a 2005 PowerBook G4 DLSD/HR and a 2006 MacBook Pro. While they look similar, none of the parts are interchangeable. The dimensions are slightly different, as the MacBook Pro is a tad wider than the PowerBook. Even if you cut down the edges of a MacBook Pro keyboard to somehow fit, the trackpad is still a different shape and would not function with the powerbook, and I doubt the keyboard would either. The displays are different shapes, the powerbook display is 60 pixels taller due to the MacBook Pro having a webcam while the PowerBook doesn’t, Your best bet is to buy parts computers off of eBay, or go down to a local tech thrift store or recycling center, as people are still getting rid of their old ones. These are fun computers to play with, and I have found that even on 10.5 Leopard they are still somewhat usable. Good luck!