Well, for #2, there are excellent modules based on the TP4056 that cost 1$. They charge lithium-ion batteries only. So check the label (yes, I know it clearly says Li-ion but don't trust strangers on the internet for safety facing stuff...), if it says Li-ion, then these are great.
For #1, probably one of those fat traces in the flexible cable is BAT+ and the other is BAT-. I would look for a way to safely remove the plastic coating in such a way that I can't accidentally short circuit it. For example, exposing the copper for BAT+ on one side, and BAT- on the other.