Hi Susan. Lots of devices only use the battery to power themselves, while plugging in, it only charges the battery it doesn’t actually power the device, so once the battery is dead, boom the device also dies, even plugged in. Assuming it’s a USB plug, you can try a larger charger, one that outputs more amps. The Speakerphone will only draw as much as it can and circuitry allows, so no guaranty that would solve your issue. Another possibly is that it’s just a weak battery and needs replacement.
Try booting again, but this time only press/hold Command and R. You should boot into internet recovery, try installing macOS on the new OWC drive, if the computer can’t “see” the drive, you’ll have to put the original drive back in and update to (at least) macOS High Sierra. I believe it has to do with the new APFS, (Apple File System) that was implemented in HS and the computers firmware. In fact, OWC should have included paper work on it, I remember mine did. Your getting the question mark folder because the computer can’t find a bootable source; which makes sense because the new OWC drive will be blank. (No OS)
I believe the thermal sensor is required on most iMac’s, 27” or 21.5” models. Since the fan is acting weird, I’d say its a good bet. I have done SSD upgrades in the past and have realized it’s a crap shoot, some iMac’s require it, while others don’t; honestly I wasn’t able to find a definitive answer especially in the 2012-2013 model years.
Since it fell, I’d say you have some hardware damage, more specifically the USB C port, even more so since you say it isn’t the first time. It will more than likely have to be changed, something probably got bent, or maybe the solder connection broke, that’s why you have to manipulate it to get it to connect. You’re probably still missing one pin/connection to actually charge the battery, almost as if you’re only getting the “DATA” connections. USB C connectors arn’t the strongest thats for sure, and to snap the cable…it definitely took a hit. It could also be an IC, controlling the charging, maybe it got shorted when the computer fell and the cable snapped. I assume you have you tried charging from both USB C ports?
I have seen it where a "shorted" or dead keyboard prevented a MacBook from booting; even using the power pads on the logic board, after the keyboard was unplugged the computer worked perfectly fine. However, since you know you spilled oil, and some keys aren't working properly, it's safe to assume that the logic board has liquid damage, and I bet it will need either a cleaning which most placed don't do, or a replacement along with a keyboard replacement,
Did any of the repair shops give a reason why they couldn’t fix it? Replacing the logic board will fix the problem, because you’re basically replacing the complete internals of the iPad. I’m assuming for the cost and time associated to replacing the board that it would be beneficial to just buy a newer iPad instead since the iPad 2 is pretty old.