Considering you haven’t had her long, you noticed anxiety issues under saddle, and now behavior expanding to ground manners…the problem is is not the gate, or the paddock. I learned the hard way that 99% of the time,behavioral problems or changes are pain induced. First and foremost, check for an ulcer. very common. Pain issues makes horses loose their head, get hypersensitive to their surroundings, and appear to us as “bad habits” that are starting. But I learned good horses do not go bad for no reason. She is hurting. My horse too got unreasonable at turn out time…started running through the gate before handler released him. Then would try running of his his stall. I believe now he was trying to run away from the pain. Could be can ulcer or many other things. Could be dental issue too. You’ll never know til you start looking.