i;d definitly reccomend a 12 gauge, with a youth stock... you can always change the stock as she grows. 12 gauge there is a much larger selection of ammo than the 20. you can get 12 gauge loads that are loaded 'down' to a 20 gauge. Recoil depends on gun weight, payload weight, and velocity. (simplistically speaking)
3" turkey loads is stout in anything. if she can handle that she can handle darn near anything IMO.
What kind of game will she be going after? semi auto will soak up some recoil for sure. heavier gun will soak up recoil, same with a nice recoil pad, and remember to use a round suitable for the game being hunted. for example no need to load up a 3" #6 shot turkey shell, when going for squirrel. where a 7/8 -> 1 1/8oz load at a lower velocity will do just fine.