I think that is too little a pot for even wax begonias. Give them more soil and some nourishment (I water mine with horse poop tea.), and I think they will take off for you.

As far as storing over the winter, I never get mine in soon enough to offer any advice there. I've got to stop buying plants that have to be dug or brought in for winter. This last winter is the first year I have ever got my cannas in before the hard frost.

I have a friend who tells me that the solution for my sloth problem is to plant the begonias in one of those hanging sacks. That is what she does. She takes it down and lays it on the ground to water it thoroughly with the hose. She also applies her liquid fertilizer like that. She has gorgous begonias and she just brings the bag in in the winter, hangs it in the basement and stops watering until March the next year. Carmellia