It is very likely that you have damaged the U2 charging logic ic chip, probably happened because you created a short on the main board while trying to replace the battery. It is repairable but it requires a board level repair.
There is several possibilities to this: A/ He's damaged the main board in his process of repair B/ Proximity or Dock flex issue (whether it be its not plugged in properly or damaged) C/ Some how he's managed to corrupt the iOS D/ TouchID flex/sensor The issue of the home button could be because he's put the metal plate behind the home button back the other way round and therefor not pushing on the bit that sticks up on the plate. My suggestion would be to go back to the repairer and get him/her to fix it, or seek some sort of compensation. **Just went back to re-read your post again a couple times, did he put a new home button in your phone? I've seen cases where new TouchID home buttons don't work together with the main board, and will go on an endless loop of restart and blue screen.
How much it costs to fix will be heavily dependent on where you are location and how much competition there is around you. I would ask around the local shops for pricing, and would recommend you request for an original screen replacement. Alternatively you could just purchase the LCD/digitizer assembly, and follow the detailed instruction on the ifixit repair section.
The first suggestion i would give is to try a new charge flex, as that is the easiest and cheapest option to try. If that doesn't fix the issue then it could be the charge IC that has been damaged (although from experience the symptoms doesn't sound right). Have you looked under all the shields with an microscope to check that there is definitely no corrosion? Sometimes not everything is visible with the naked eyes. You could probably try an isoprop scrub down anyway, you don't really have much to lose at this point in time.
You may also want to check that the plug is all the way in, or the socket to the LCD cable is not damaged in anyway. I've had experience where the LCD is perfectly undamaged but the socket where the LCD cable plugs in have been bent a bit out of shape and its not making full contact in turn it created an line down the LCD approximately an inch thick. Once we manipulated the socket it went back to normal again.