Brand new batteries? From a reliable supplier? If you have a universal charger available you could manually boost the batteries back up for further tests. That being said it sounds like your logic board is shorting out, are there any hotspots on the board? Externally charge up a battery until it is full, and get the Logic board nice and cool. then plug in the battery and lightly feel around and see if there is any spots that get particularly warm. If you have a board to compare it with check the resistors around the LCD, Digi, and Prox sensor connectors. Peel back that little black sticker, it is often those that I see people nicking but it is odd that it wouldn't cause a total failure to boot. This is also an OEM cable and OEM block correct?
Not very easily and not without at least a heatgun, hotplate, pry tools, a can of compressed air, steady hands and a bit of luck if its your first time. It is very likely you will also need replacement adhesive and a new charging port. This is NOT something I would attempt for the first time if there is still hope of saving it.
This sounds like a problem with the software compatibility of the apps you have named, your best bet is probably to send in a bug report to every app that you are having issue with and possibly HTC. Until then you may have to use the older versions of the apps until a patch for your issue is released.
Check your battery tab in settings and see if anything is draining an exceptional amount of power. if nothing seems out of the ordinary backup and DFU your phone. Before restoring see if an unloaded stock OS still produces these issues. that being said, it sounds like your phone may have suffered water damage and I would get it looked at ASAP.
Have you tried factory restoring the device with the original home button? (this could be very risky if the button is damaged or it is not connecting properly.) Are there any signs of water damage in the device? Also how do all the connections look inside? Have you tried cleaning the connections with a ESD safe brush?
Assuming the part is 100% working, which I would not without testing it on another device. Clean the contacts for the LCD and board with a ESD safe brush that is slightly damp with 99% isopropanol. This could be a poor connection.