There are 3 sizes: S, M and L, the larger ones usually have better stats but consume more SP per turn, thus needing to drain Kuro more often.
Also, the effectiveness of some skills depend on the sizes, for example, some L-sized girls learn Stomp, which has a chance to insta-kill S-sized girls.
Besides the passive skill that rises some stats while in skinship that every girl has, it also helps build Affection quicker. Keep in mind you can change the size and transparency of the skinship animation in the Settings, or by pressing Q.
Partially answered in question 1. The amount of SP they lose per turn is a percent, so increasing the max stamina won't reduce the frequency they need to replenish it. If you don't want to spend too many turns draining Kuro stick to S-sized girls. Remember you can recover it with certain items too.