My issues started a month and a half ago. At first iTunes just didn't recognize my phone as being plugged in. That sucked, but I could still charge it using the AC adapter. A week or two ago, I noticed that the phone didn't automatically recognize being plugged into the adapter, I would have to press the sleep/wake button then after a couple seconds it would recognize and start charging. Then, today I noticed my battery getting really low, down to 3%, so I plugged it in and nothing happened. A short while later I got the dreaded white screen error. I had experienced this before, but now my home button is broke so I can't do a hard reset or DFU mode either. I again tried plugging it into my computer and nothing really happened. After 10-15 minutes it started vibrating as though it was ringing. I couldn't answer it, so I unplugged it. The vibrating stopped and the screen went black, it turned off I assume. I plugged it back into the computer and it turned on and now the battery read 23%. The only thing I can think of is that it charged while it had the white screen for some reason. I know this is a long post, but I didn't know what info would help someone figure this out.

In summation:

It's not the cable(s), I've tried two, they both work with my iPod, but not my iPhone.

The phone isn't recognized by the computer, or iTunes, but if it is off and then plugged into the computer it will turn itself on.

My home button doesn't work.

iPhone 3gs, version 3.1.3, MC model
Windows Vista, service pack 2
Lastest version of iTunes, reinstalled it today, along with Apple Application Support, Bonjour, Quicktime, and Apple Mobile Device Support

One of the suggestions I saw on other forums was to try using DFU mode, I saw that it might be possible to do that with a jailbreak, but first I need to get my phone to be recognized and to charge.

More often than not, this type of error is hardware related. There are only a few things to check and it looks like you've already checked, but to recap:

1) Make sure you've tried another Sync Cable.
2) Make sure you've tried another USB port on your computer.
3) Make sure you've tried another computer.
4) Make sure you've tried connecting directly to a power brick.

That said, it looks very much like you've tried all that. I'm not one who believes that restoring the software can have much effect on this, but the issue you're does seem like it could be software related as the phone would charge properly when the device was plugged in after being completely turned off.

If you want to force it into DFU mode and install 3.1.3 (the only thing you can do unless you have an SHSH on file for 3.1.2, which will LOCK your phone until July when a new unlock is released), you can do that -- use the forum guides on entering DFU mode to get to it, and complete the upgrade. That said, only upgrade to 3.1.3 if you are on a legitimate carrier with a legitimate SIM -- you can jailbreak 3.1.3 with Spirit, but you cannot unlock it under any circumstance at present.