If two eggs were fertilized and started to develop, that is pretty good evidence that you have a male/female pair. (Parthenogenesis in bird eggs can happen but is pretty rare: that would be when an egg starts to develop without being fertilized.)
I don't have a good explanation for why there would be so many eggs if only one bird is female-- maybe she's weird? If both are female, then the egg count is a little bit high but not too surprising.
Yes, she would move her bum around when "mating," regardless of whether her partner is a male or a female.