Try putting it in DFU mode. Hold down the home button and then plug it into the USB cord attached to your computer with iTunes up. That should kick it into DFU mode and iTunes will detect that it needs to be reset and will go through the steps of restoring it again. Hopefully that'll fix it.
The white screen, also known as white screen of death, can be caused by a couple of things after a fall. If you're lucky all that happened was that the LCD connector came disconnected and you just need to lift the screen up and securely connect it again. If you're not so lucky it means that your LCD is now not working. ifixit has a step by step for replacing the LCD. Try following steps 1-4 except when you get to step 4 try pushing that connector back where it belongs instead of unattaching it. Front Panel Assembly Instructions Good luck! Let me know what happens.
I did this replacement also and on the power button at least I found out that the metal loop attached to the power button has to be down for it to work correctly. I put it up and it only stuck out about half as far.
It really isn't necessarily a hardware problem. Try getting the phone into DFU mode (hold down home button and then plug into USB plugged into computer) and restore it. If that doesn't work there's another thing you can try but it's kind of complicated. Let me know if it works.
Connector 1 which is a pain in the you know where to get seated is more than likely what is giving you the problem here. It's hard to tell if it is completely seated or not. I would suggest pulling the screen off again and checking connector one to make sure it's fully seated in there. You might also want to try letting ifixit know that the LCD is not working they might send you a replacement to test out and see if the product is faulty.
I've done this repair and it's a lot of work. You have to open the iPhone from the back to replace it. While you can buy the whole display assembly I did it with just purchasing the touch digitizer. It's a little more difficult but will save you $40. Shop around on eBay for the touch screen. Don't buy from China on this one. Read through the step by step on how to remove the display assembly and then check out youtube videos on how to separate the touch digitizer from the main display assembly. You're gonna need a hair dryer and one of the plastic pry tools. You can do it. I did that as the first thing I'd ever done on an iPhone. Take it slow when disassembling don't rush things. If something isn't coming easy don't pull harder. I did and tore a ribbon cable. Best of luck!
I had this problem. While this may not make any sense put the suction cup near the home button and when you get ready to pull up also push down on it. Don't push down so hard that it isn't going to want to pull up but apply some pressure. This will usually keep the touch screen and the display assembly attached. If that doesn't work try using a blow dryer on low heat at about 6 inches to a foot away and heat up the bottom portion of the touch screen for 10-30 seconds. Then hold the corners of the touch screen down for about a minute or so. Let that sit then for a while and by then the adhesive should have set and you can try using the suction cup again.
Water and electronics usually don't mix well at all. I'm working on a 3g that was dropped in a puddle by a guy out hunting (why you would take it hunting I'm not entirely sure). Fairly often the LCD needs to be replaced when there is water damage (that's why the screen is black). Sounds like the speaker is having some problems. The speaker is contained in the dock connector assembly at the bottom of the iPhone. Plugging the headphones in shouldn't require the phone to be restarted. If you look into the headphone port there's a liquid detector in there. It'll turn to a different color if there's significant water effects. The fact that it won't play music out of the headphone jack probably means that it's gone. Does the volume up and down and vibrator switch work fine? Those are connected to the headphone jack and might narrow down if the headphone jack is ruined or not.
I agree it does sound like you need a new touch digitizer. These can be found for relatively cheap ($10-$20) on eBay. The only problem with just replacing the digitizer is that there is a frame that the digitizer sits in/is attached to and that frame has the home button on it. Buying the whole assembly is about $60 on eBay. There is a step-by-step on here that shows you how to replace just the touch digitizer. I have followed this step by step and successfully replaced just the touch digitizer. My only warning is that you need to be really careful not to bend the frame that you'll be removing from the old digitizer. I bent the frame a little and the touch digitizer didn't fit in as well. Good luck I'm sure you can handle it!