You can but it is better to train its EVS before leveling up to reseve berries.
You can, it is good to not evolve a pokemon at a low level, only be careful from errors when you are near 100 lvl.
Beating the e4 and the champion give you acces to the next region.
The level or the pokemon used is not important only you must win.
Cerulean cave is a quest with its requirements. (Beat Lance in Dragon Den/Johto to get acces to the cave).