I have a Solenia Dusty rose begonia and noticed it is an annual plant. This is in my apartment (only place to grow plants on 3rd floor. Can I keep it growing in my apartment?
Yes. This article will help: https://www.gardeningknowhow.com/ornamental/flowers/begonia/begonias.htm
Can I save them a bit? In front ok (southern exposure) just watered those! Back (north) in pots on a wall nipped pretty good, have not added water. Tonight prediction is not as cold as last night.
This article may help: https://www.gardeningknowhow.com/plant-problems/environmental/tips-for-saving-cold-damaged-plants.htm
I would move the plants, if possible, to a more protected area.
This article will help you: https://www.gardeningknowhow.com/ornamental/flowers/begonia/wintering-begonias-overwintering-a-begonia-in-cold-climates.htm
Someone gave me a starter and I am not sure what kind of pot I should use and what I should expect in the way it will grow and look.
This article will help with growing begonia plants indoors: https://www.gardeningknowhow.com/ornamental/flowers/begonia/begonias.htm
I use a "professional" 20-10-20 water soluable fertilizer on all begonias. You should be able to buy this from your local greenhouse.
This article should also help: https://www.gardeningknowhow.com/plant-problems/disease/treating-root-rot-gardening-tips-for-housplants.htm
If the plant is rotting, it is probably too wet. This article should help: https://www.gardeningknowhow.com/ornamental/flowers/begonia/begonias.htm
Begonia corms were dried and stored in the garage over the winter. I went to lift and divide them and discovered white grubs had eaten them; hence they are rotten. What went wrong?
Hopefully, this article will help answer your questions: https://www.gardeningknowhow.com/ornamental/flowers/begonia/wintering-begonias-overwintering-a-begonia-in-cold-climates.htm