Yes, heating the silver up to the temperatures needed to flow silver solders is enough to anneal the silver. To say it another way, it's enough to remove the temper or springiness of the silver. .....particularly if you let it air cool.
I think it is less of a problem if you quench the hot piece in a pickle pot. I'd have to get out my books to verify this but from experience (in the distant past) dropping the hot piece in the crock pot full of heated pickle solution left the piece with at least some of the temper.
In practice, annealing a jump ring is not a problem. The soldered small round or oval jumpring is now a closed loop and not very easy to deform even if it is dead soft. An ear wire is more of a problem. You are working on one end of it. The unworked plain piece of wire that goes thru the ear can get so soft that it is too easy to bend out of shape. For that hammering may be the only way to work harden it.
I asked our usually silent co-owner, Dee Mueller, how she kept the temper in her pieces. Dee says she'd use the micro torch with a very small, very hot flame and try to only let the end of the ear wire get to any color from the heat. If she was melting a ball at the end or soldering a fancy end on the wire she'd try to keep the rest of the wire from heating up too much. And she'd throw it in the pickle as quickly as possible.
By the way, getting that ball at the end is never as smooth as you'd think it should be. It usually is a little off center but that usually doesn't show once you bend the tip a little and assemble your earring.