Yes you can - my mum has one which is about 30 years old and has moved house on more than one occasion during that time. The last time it was moved was about 7 years ago - it's still dong really well. I'll see if she can send me a photo of it. It's a beautiful deep pink and flowers profusely. It originally belonged to my grandad who died in 1987.

Yes you can move a peony but don't expect many flowers for a year or two as it takes that long for them to settle in.

They hate being moved so be prepared to loose it? Having said that I moved a very young one last autumn and it is sprouting.