I would be more worried about ensuring the voltages between the batteries are virtually identical. The voltages need to be close to in order to avoid any problems with over charging which can lead to fires that are extremely difficult to put out. So if both of your batteries read 3.7V or one reads 3.6 and the other reads 3.7, you should be fine. You just don't want them too far apart in terms of voltage.
To address your original question, if both batteries are already fully charged, then there's no need to charge them. They should right around 4.2V when fully charged, usually slightly under. If the voltage there is a bit lower, it'll be fine to use them in the build for testing so long as you keep the cells balanced (meaning you keep the voltage roughly the same as I stated above). But it probably wouldn't hurt to fully charge the unit after you finish assembly.
Yeah, you're gonna want to use 22awg for the batteries as well as for things like the charging wires, and the power wires to the Wii and screen. Anything that draws a good amount of power you'll want 22awg for.