ya not pretty but it should be fine

if the flipper is stuck on the transistor probably failed but not necessarily from the hack

I would trust a piece of wire (as used) more than a trace pen personally

whatever solder you use should have rosin core so you should not really need flux for any of that work... you might want some pcb cleaner/flux remover though to clean up after

that depends on the tools being used... if you are using braid then yes, flux is almost mandatory... if you are a using a nice de-soldering station with a pump it is unnecessary

If a part is being uncooperative I usually just add a bit more solder as you mentioned (I just use the desoldering gun itself so not much reheating) so like I said, its unnecessary, as in, you don't HAVE to... but you certainly can use flux if you want to go that route... like I said when I am working on large things and need to use braid flux is like magic... different strokes