I have 19 Tilapia in a 55 gallon system that I've had since fingerlings. That was 6 weeks ago. Now some of them are getting larger, but not all of them. Some are over two inches while some are about an inch. Perhaps the smaller ones are females? These were mixed sexes intentionally.
Also, one fish seems to be a loner, while the others stay fairly close to one another (usually near the bottom of the tank).
Plan on outgrowing that tank in the next month or two.
Tilapia behave much the same as all Cichlids...they will chase each other around, nip at each other, get territorial, etc. They also eat and grow at different rates allowing bullies to emerge. The size variations you see now are probably not sex related.
Nothing unusual in what you've described. You really need to look at getting a larger tank cycled and ready ahead of time (I'd use some of these fish for that as well).