The amount of time varies too. I don't play that long in one sitting but if you've got the gamepad screen's brightness turned up, volume up, and its displaying something other than a stationary map, you can be sure you'll see that red light crying out soon. It usually lasts me a good 2.5 hours before it needs a charge.
Hmm, that's a tough one when they are small like that. I haven't run downstairs to check out the screw you're referring to but are you able to use a slightly larger screwdriver? Sometimes that works. Sometimes using a rubber band over the screw adds enough grip. I've even tried super glue to add grip but I wouldn't do that with electronics, only wood.
I think you're going about it the wrong way though. I don't own a Wii U, but I imagine I wouldn't want to have to charge it after/before each session. That would be like having to change the batteries on your remote every time you watch TV. It is an inconvenience that could have been easily remedied with a bigger battery, specially considering that the compartment inside the Gamepad is bigger than what the current battery holds.
Anyway, I hope they make this standard moving forward. Perhaps by next Christmas they'll do so just like they eventually put in Wii Motion Plus on all Wii Remotes.