lithium ion battery queries
Yes, this connector should fit the one of the battery.

Also, all Lithium rechargeable batteries suffer quite a lot from voltage drop. This should not be an issue, since for the motors the voltage is not critical, and the logic stuff uses regulated 5V anyway. Li-Ion and Lipo are the same thing. The only difference is that the term Lipo is usually applied to batteries in a plastic pouch, whereas cylindrical  cells are branded Li-Ion.