The problem with trying to build a connector with a balance plug is finding a housing that holds that exact size battery. If you could find a battery tray that holds 10400 batteries then wiring it up to charge 4 in series is pretty easy.
Yes
All lithium batteries can be charged at a 1C rate. So for a 550mAh battery youd want to charge at about .5 or .6 amps.
Correct but dont charge them that way. You'll either need to charge them 1 at a time or in series with a 4S balance plug connected to the charger.
No worries. Happy to...
Theyre going to charge up to 4.2v but 1.2v isnt likely to hurt anything. Most electronics have a tolerance of a few volts above what they are actually rated for. That being said I can't make any guarantees as I've never done it myself but if it were me, I wouldnt be afraid to give it a shot. Then...
Thats hard to say. I'm not familair with the xbox controllers. It looks like they use a special battery pack. Youd probably have to take the battery pack apart and see what kind of cells are inside of it and see whether or not those cells would fit inside the proprietary case.