Well--all begonias aren't the same, and I don't know of any begonias that are annuals. Some are grown as annuals, or as cut flowers more or less, to throw away when people get tired of them or don't know how to grow them for longer periods, but so far as I know none are annuals. As far as growing rexes outdoors, while your conditions are somewhat different from mine I would think you could grow them outside just fine after frost chances are over, as you say; I grow most of mine outdoors. Be sure they're in shade and in a cooler part of the yard--they don't care much for heat. If you're interested in learning about the different kinds of begonias, check out the AMerican Begonia site. Too few of us get to see all the different kinds. They just aren't available most places, and it bothers me that the ones that tend to be most often available are often the ones that are more challenging to grow. I guess a lot of people in the business go by the motto of someone local, who fortunately isn't in the business anymore. He said sell people plants that will die so they'll have to come back for more. I think he went a long ways to convincing a lot of people they had 'black thumbs.'

Dear greenelbows,
I'm so sorry that I haven't thanked you for your advice, it was very helpful. Recently I've been ill so I haven't been getting back to people as I should.
Thanks again,
~ Georgia

No harm done, Georgia. Hope you're all better now. Illness makes it hard to do what we expect to be able to do, even if, as in my case, it's someone else who is ill (my husband.) Be well, and enjoy your begonias!
Nancy