My brain doesn't seem to be in very good working order tonite, but maybe I can get things started and hope others will pitch in. If they don't, you could try the American Begonia Society website (ABS.org) and see what you can find. They seem to have re-organized it lately, and I had trouble finding my favorite site for making identifications on the Astro Branch site (linked from the ABS) the last time I checked, but maybe it's back. Anyway--#1 is a Rex hybrid, but there are a number of them that look a lot alike, and then there are lots that are seed-grown and unnamed. Don't know #2. #3 is B. partita or one of that complex--they are somewhat confused and I think I read in the process of being untangled. #4 is Mirage. Don't know #5 but I'd guess it is a ricinifolia hybrid--very nice. #6 might be Tiger Kitten but I'm not sure. #7 I should know but I can't remember--maybe I can find it. #8 is gorgeous and I hope you find a name 'cause I want it! #9 is like #6--one I'm pretty sure I had but a while ago and not for long, maybe Persian Brocade. (I love begonias and tend to get too many when I go someplace where they have a number of them, and invariably lose a few that either don't like it here or don't get enough care.) Good Luck!
Nancy

Greenelbows, thank you for your reply!