1. Not that i know of. And if there were then that wouldn't be fair. Buzzerbeater has clearly stated that Supporter feature will not give an advantage to the users who have it so there is no reward or money. I'm pretty sure there is no trophy either. Private leagues are mainly just for the fun of it, bragging rights, and for if you want to see what your team would look like in a playoff game if you played them "this way" or "that way".
2. Private leagues start and end all at different times and it does not go along with the schedule for the league play. This is because there are different lengths for leagues according to how many teams they have, what their playoffs look like and how many times they play each other.
3. I'm not sure exactly what your asking but if you are asking how having a private league will effect your leagues games/ the cup, it wont. Private league games go into arcade mode so the minutes and injuries that occur during private league games do not matter for your training or game shape or injuries.
Hope this helped
E.B.W.
Murray/Harris/MPJ/Grant/Jokic - 2020 NBA Champs