Do you mean Clan? You can’t form a new faction, so far as I’m aware.
If you change Clans or form a new one, then it won’t affect the academy or faction flag bonuses. You get those just for being in the faction. Clans are just a sub-set of the faction, so you can change Clans without changing faction.
There isn’t a waiting period between leaving a Clan and joining a new one, although you have to be accepted to the new one.