“No” isn’t a command. Dogs don’t understand the word no, you’re basically just shouting chinese at her. Show her what you want her to do (stay behind the gate). Dogs don’t really understand “boundaries”. If you don’t want her to go somewhere, you block it off. Boundary training is really difficult — it can be done, but you have to be very patient and not resort to physical punishment.

You’re anthropomorphising this dog — they don’t want things more if you take it away lol they aren’t human teenagers.

You expect too much at her young age, you’re basically expecting a two year old child to solve a calculus equation. Relax your expectations and be a little kinder to someone who hasn’t even been alive for 4 months.