First thing, does it show up on any other device? If it isn't showing up on any other devices, then your hotspot isn't broadcasting a signal. Otherwise, it could be broadcasting a signal but your HP might be getting cross talk, is too far away, has one of it's antenna's disconnect, etc.

The external PSU shouldn't harm your system. However, you'll need to have a way to turn on the external PSU. Usually a paperclip between the green and black connection to power it on.

There's a potential that it will. "Potential" It could be ok, but any time there's a burn mark on any sort of adapter, charger, plug-in, you're only asking for trouble in the long run. If it's as easy to replace as that is, replace it.

So are you saying it works until you do the Time Machine Reload? If so, you might have something corrupted or some bad software in that back-up.