[Motorola Droid 4] RyanV-M:
UPDATE: Got it to power on and boot normally once. Enabled USB debugging when I had the chance.

Got back from the cell phone repair place this morning. He said the battery was definitely bad, drawing zero amps. He also said he was getting weird voltage spikes throughout the board, and I think he said something about the PCMI power management chip being bad and that the phone was done for. I told him that I could plug the phone into the the computer and it recognized it. He said my best bet might be getting a battery off eBay that would power my phone long enough to get everything off of there. I asked if it could also be USB cord related as I tend to have horrible luck with them. He said it could be and that the reason that I was able to have the phone recognized by USB but not charge is due to different pins communicating during USB connection. I'm including these details so you can gauge this guy's expertise. He did go from saying the phone was done to suggesting a new battery/USB cord, and I was able to power it on after using said cord.

I went and bought a new USB cord and plugged it into my inverter I had in my car. To my surprise after a few seconds, the phone was showing the charge screen with the battery at 70%. It still wouldn't boot up so I kept it plugged in for about 5-10 mins. Eventually it did boot up. I immediately went in and enabled usb debugging. All my data was there. For a brief bit the date and time was off, but even that corrected itself. I played a song from my music collection on my SD card to make sure that was OK. The song played, and shortly thereafter it looks like the phone powered back off as it had originally done. I hooked it up to my computer and my PC started installing new hardware, it said something about xt894(my phone) a couple of disk drives and a CD ROM drive, and something about adb. The furthest they got were as unknown devices, so I disconnected them and reconnected them a couple of times, eventually the install icon went away. It still recognizes the phone and even recognizes there is a storage device with the phone in the device manager menu, but it still says "please insert a disc into drive (letter)" The drive with just motocast is still there, and I think there was another CD ROM that was recognized, said to insert a disc same as the drives. I threw it on my wall charger for a few minutes, but it still wasn't powering on. I then realized that charging it could exacerbate the voltage spikes the guy was talking about, and I didn't want to risk it especially after seeing all my data there, so I unplugged it.

I do have the battery that the extended battery replaced. It had to be pretty much tied to a charger, but it did work if only for less than half an hour. So there is the option of using that to temporarily power the phone.

My new questions are:

Am I at risk of further board damage if I charge/power on the phone? What about charging the degraded but functioning battery?

Now that I know I have USB debugging enabled and my computer still recognizes my phone, how do I go about backing up the contents of my internal storage?

Could this all be effectively solved by just getting a new battery? What about what he was saying with the weird voltage spikes and PCMI chip?