Some people don't read before they post-
Great job on getting her started, though the skeet episode could have started smoother-you owe that gentlrman with the 20 ga. a big debt.
It's great she wants to go bird hunting, and a 20 ga. gas-operated gun is an excellent choice. Offer to show her how to clean and maintain it, and point her towards a good gunsmith for when she doesn't have to time or just doesn't feel like it. (You should do this even if you offer to clean it for her, you may not always be there to do so.)
Don't forget the NRA's instructional books-I just gave the NRA Rifle Marksmanship book to a friend's daughter, she comes out and shoots some of our .22's and other guns (She's even shot a Mosin Nagant and AK-47!
) when we shoot at their farm. She might even let her Dad look at it when she's done!
My own opinion as to an HD firearm. I keep a .38 Spl. as my first line of defense, the one my wife can 'grab and go.' I have an 870 at the ready, she has been shown how to use it, doesn't like long arms however, and won't shoot it. That's the one I'LL grab while handing her the Cobra.