I'm not sure of your scenario, I think it could also have been improper connection. The screen uses about .2A at 5v so the wire thickness is not crucial, there is no difference.
The tube goes in long end and it will insert to be flush with outside of the case. You may need to open the hole a little with an xacto blade (from inside insertion). It is suppose to be very tight, it is possible.
Make sure the batteries are fully charged (up to 4.2v). 2.9v is dead battery and it cannot possibly power the wii, not sure if you charged it since initial message.
First observation is that you should not be using thick wire for the small solder pads. You can use the magnet wire for the connections to the USBc board and the rgb led. Also you can clean the burnt flux residue with isopropyl alcohol. Check if the usb data connections from the wii are...
I cant really say for certain without feeling it myself. You just need to use your best judgement to figure out where the button is rubbing and getting stuck.
You might need to file a little material off of the shoulder button rotating hinge part that pivots against the pcb. Only take a little off at a time to avoid mistake of taking too much off.
Yes they are the same. You don't need a lot of airflow for this rework, I keep the air flow pretty low.
I see lots of solder blobs on U7, jobs not done. Do not apply power to it until it looks perfect.