Welcome to PokeMMO!!
1) Trainers are a 6 hour cooldown and Gyms (including Morimoto?) are 18 hours. These are hours in real life and not in game. Each in game day is 6 hours.
2) I believe Morimoto is easier the first time around. When I do rematches with 6 lvl 100s, gyms and Morimoto are around lvl 80-85. They could be scaled to what you bring to the fight, I'm not sure.
3) Battle points can be seen on your trainer card.(Second icon on the bottom or press c) They can be used at the Battle Subway(Unova) or Battle Frontier(Hoenn) to purchase items, certain moves, etc.
4) In PokeMMO eggs hatch in 10 mins or 5mins with Flame body/Magma Armor ability. Shiny rate is 1/30000 without any modifiers. So with the donator pass it would be 1/27000. 10% off
Enjoy your stay! :)