Carefully inspect the LCD flex that connects to the board, or in this case I believe it connects to the charge port flex. Try applying a little pressure to it or check for any signs of laceration, if the flex is torn at all then the LCD is no longer good. I make the comment about the pressure because something similar happened with an LG G2 i fixed. No image on the LCD but when I would play with the flex the image would show.
Sounds like an antenna issue. Open it up and make sure the white antenna cable is not loose and that it is the original one to that phone model. I had a similar instance with an S3 that was a TMobile T999 but had the Sprint L720 antenna and the signal would come and go.
There are two possible fixes. 1 - This method will take trial and error and is not guaranteed to work, but it will be by far the easiest solution. Try to find a thick tape and maybe a foam spacer and place it where the arm bracket should be also make sure its applying a little pressure on the component with the LCD ribbon inside. This has work for me many times on Blackberry's and HTC Evos. 2 - This is the legitimate method, guaranteed to work (if done right), but has a very high level of experience needed. You have to buy the whole connector that goes mounted onto the board. Unsolder the broken one and solder the new connector. This part has A LOT of pins, unlike a charging port or power button so diligence and patience will be a virtue. If you don't have the equipment, I would order the parts (which are maybe $3-$7 but can be hard to find) and take them to a local repair shop that does mother board repairs (since its unlikely they will carry that in their inventory).
Yeah of course, it's not a ribbon or flex that you can unplug and plug back in, its simply going to need to need to have the power button re-soldered or replaced. If you don't have the the equipment or knowledge on motherboard repairs I would take it to a local repair shop since this is delicate work that has to be done directly on the board, and a wrong move can cause serious complications.
I'm 80% sure it's the battery, because batteries have fuses in them that like all other fuses, they "go out", so the battery is receiving charge but isn't storing any. In very rare instances it would be a component on the board that regulates the electric flow into the device, but honestly I think with a new battery you'll be fine.
notice when separating the glass from the LCD that there is a blueish ribbon on top of the LCD. That's the digitizer itself, if you slice even the most minimal part of that ribbon the touch will not work anymore.
There's a spray called "contact cleaner", used mainly for cleaning connectors on boards, which is fairly inexpensive but highly effective. I use it to remove any glue residue off the LCD's then use an LCD cleaner for shine. Until now I've never had any problems!