"Is too many live-in boyfriends and moving too much a good reason to prove my ex-wife provides my children with an unstable environment?"
Depends on (a) what "too many" means, (b) what "too much" means, (c) whom you're trying to prove it to, (d) the standard of proof that person uses, and (e) why you're trying to prove it.
"Could I fight for the majority of Custody?"
You can fight for anything you want. You can certainly make an argument that your ex's proclivities are not in the children's best interests. No one here can possible predict how this might turn out.
"When I have my kids we all share a room, but the room is ours! Would the courts look at this in a negative way?"
Yes. A parent and two or more children in a room is not a particularly good situation. The court may view this as a choice between two situations that are bad for different reasons. And, keep in mind that the cost of having children is only going to increase if you have primary custody.