Don't worry, just keep doing what you're doing and things will be fine. Little ones seem to be like the Tasmanian Devil (remember that cartoon
?) She'll calm down as she grows. Keep trying to wear her out
About the gate, try to find one that has either very small holes where she can't get a foothold to climb, or something with close-together vertical only bars. Sounds like you have a little escape artist on your hands
Hang in there, Mom, and good luck with your move
Welcome to YT!!!!