I don't think they'll have problems fruiting in somewhere like Georgia, zone 9. In the South, with its humid summers, the issue will be disease. Will probably require a lot of spraying and your cherry tree (Prunus avium I'm talking about) may not last too long. Insects may also likely be an issue. There's a reason cherry trees are not traditionally grown in the South.
In case you were interested, native Black Cherries have slightly more resistance to the disease and insects.
Late spring freezes can also be a problem with stone fruit in many parts of the South. Will zap those new blossoms before they even have a chance to start growing fruit. You should still be able to get fruit some years though. Cherries tend to be later to bloom than peach by maybe about 2 weeks.
Oh, by the way, it is physically possible for any cherry to pollinate any other cherry, regardless of what specific species it is, as long as it's a different cultivar variety. (There's only a tiny number of exceptions, because a few cherry varieties may be too closely genetically related to be able to pollinate each other, for example the cultivars Bing, Royal Ann, and Lambert)