If you need to hold the USB cable in a particular position to charge, then it is most likely physical damage. But, I had this happen for one of my devices. I found a little bit of paper or something had gotten stuck in the port. Once I cleaned it, everything was OK.
As to why only one charger works, if it isn't a fault in the socket, I don't know. But, does it charge when you connect to a PC? Does it connect properly?
The problems with power-off have been mentioned in other threads. It's not something I use, so I haven't noticed it.
If a factory reset is needed, there are a couple of things you can do to save your reading status. If you use calibre, you can use my Kobo Utilities plugin to store the reading status in the calibre library, and then restore it later. Alternatively, you can take a copy of everything on the visible partition when connected to the PC and restore that after the factory rest.
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