ladyfreckles
Acrobat
yes, leave the door open (trust me you won't win the battle of a closed door).. and doggie doors (if you are referring to the dog gates metal or plastic) won't keep them out they will climb over them or jump.
I have had cats (since I was a kid) for years and I have never had one that didn't mind a closed door. All of them scratched at till it was opened.
No doggie door as in the flap animals can go through when a door is closed.