YES and YES. I have a patterned hen with the blue gene (which I don't want anyways) that has learned that if it jumps, flaps, and grabs the chicken wire, it can scale a 6' fence and join the littles in their grow-out run - where it terrifies them, then eats all the expensive 24% protein feed. It also tends to spill their waterers.
I'm culling it before it teaches others.
The rabbits are in the same space, sometimes it will hop into their runs, and eat their pellets, too.
Good night! Don't you just love having a bird like that? It's just the best. I hope she makes a good stock or sausage for you.