You haven't specified how tiny the hive is, but if it under 5 frames of bees, they are going to have a hard time growing. It's hard for them to thermoregulate or keep their brood at 95 degrees constantly. It's also hard for them to send out foragers as they are occupied taking care of the brood (feeding & keeping at 95 degrees). To help them out and get them to grow, I'd recommend two things: an insulation board in the top (to prevent heat from escaping above) and feeding bits of pollen patty (1 inch x 1 inch pieces) every week. The pollen is to feed the brood. If you want them to build out wax, you'd want to feed them 1:1 sugar water too, but that can attract robbing, so make sure your entrance is very small (1/2 inch wide). Lastly, you could get a Bee Boost for the hive (nurse bees and a frame of brood that's hatching out) to help their population.
I think supporting the hive in that way will get it to grow. When they are so small, they need some help. I wouldn't worry about the queen laying two eggs per cell. Better two than none. I trust the bees to replace the queen if they think she's a problem. Good luck! -Jennifer