My riding instructor bred her 18yrs mare for the first time last spring and she caught and is still in foal. Just as in humans, its true, as you get older the less likley it will be to have offspring. But I think she still has a few more years of producion left.
As for what breed you want it all depends. You said your not interested in selling the foal which does in a way open alot of doors. Is your mare registered? Perhaps breeding to a TB isnt a bad idea? Im not sure which traits you would like to bring out of your mare, but remember its all a game of chance, as im sure you already know. Not to be sterotypical, but arabians are generally a "hot" breed as are TBs. Perhaps breeding to one is a bit of a risk. A QH may ad a bit more bulk in the foal, but it all depends on what you are breeding for.
You said you wanted to breed your mare because of her personality? Well generally the foals attitude is developed when outside in the world with its dam, although it never hurts to try to equal the personalities between both sire and dam. Are you also trying to obtain a certain build? Is your TB mare slender or bulky?
I hope I helped a little bit, Im use to breeding standardbred and bascially its whoever has the most money, fastest time, and best offspring that we breed to. When you choose a stallion try to find foals that have dams of the same breed as your dam.