A) Yes, but very slowly. You also need a charger module designed for LiPo batteries to prevent overcharging, and a protection circuit to prevent overdischarging. Otherwise you will destroy the battery very quickly.
NiMH cells are much easier to use and tolerant of abuse. They can be trickle charged by a solar cell indefinitely.
B) the ATMega processor on the Pro Mini will run at any voltage between about 2V and 5.5V. No regulator is needed, but other circuitry might. There is a regulator on the 3.3V Pro Mini and it might be damaged if you connect the battery to the Vcc pin, in an attempt to bypass the regulator.