I can be a little help. Not all varieties of begonias are grown from tubers. None of mine are, because they don't grow well where I am. Most of mine are cane begonias. They can get several feet tall and have clusters of blooms.

Here is a link that might be useful: cane begonia article

Fall can be a good time to transplant, it depends on your climate, I would guess Maryland is too cold for begonia tubers to be left in the ground over winter. b. You have described a good way to transplant. c. I wouldn't fertilize any plant in the winter, there would be no growth to benefit it.

What sort do you have? If you definitely have tuberous begonias and they've miraculously survived a Baltimore winter; then you've got a good warm location for the original group and you're in good shape to experiment. You could transplant a chunk (def. don't feed and don't water much), bring some tubers in, and transplant some into pots and let them do dormant inside.

Here is a link that might be useful: overwintering tuberous