I have to admit, when you said something about having another horse for your girl to play with, I was wondering just what kind of "play" you were expecting. I have only mares here, but I still see some of this kind of behavior, even the mounting.
First of all, having a stallion/gelding mounting her will not cause a mare to go into heat - that is something that her body will do, regardless of the presence/absence of a male. Having a male around simply encourages the females to show obvious signs of what their hormones are doing, it doesn't change the hormonal cycle itself - the myth of "bringing a mare into heat" is just that - a myth. Just how a mare behaves during this time varies - some are really obnoxious, and some show little in the way of behavioral changes. Honestly, at the barn where I worked, we had more trouble with the geldings when the girls were cycling than with the girls themselves - the flirty hussies kept the guys so riled up, they were constantly beating each other up. Someone once told me that the only real difference between a stallion and a gelding is that a stallion can settle mares, and in some cases, I swear it's true. While there are a few geldings that seem to be "dead from the waist down" as they say, most still know that they are male, and some can (and will!) go through all of the motions, given the opportunity.
I don't know where this idea that "mares run the herd" comes from, but what I have seen doesn't bear it out. Dominant horses run the herd, and they are just as likely to be male as not. I watched in astonishment as a gelding that had formerly been the bottom horse in every group he had been put in for years suddenly decided that he was a stud when some Appaloosa mares got put into the pasture next to him, and he beat the stuffin' out of all three of his current pasture mates (including my mare, Latte, who had been the top horse at that point). A few months later, even he got bumped back down the ladder when a very dominant Arabian gelding got put back in the pasture and reclaimed his former spot at the top.
Older horses that are in good health usually dominate immature animals, so I'm betting that this gelding outranks your filly. Obviously, you don't want her getting hurt, so I think your choice to separate them is the best thing to do at this time.
Personally, I can't think of a worse reason for breeding a growing youngster than to prevent her from going into estrus, so I think that "option" shouldn't be an option at all, but should be put right off the table.
This is something you will need to talk to your vet about. The cost and risk of the procedure seems to depend on how it is done, and there are apparently a number of ways of doing it.