4 months is not long in a cat's life, my cat's took a year to get along, As fro her not interacting, most females are not the most social, they prefer to stand back and observe. They are the rule enforcers and the disciplinarians. Can you approach her and touch her at all? Try to pet her when she is eating a good treat, offer shaved deli turkey or tuna. She needs more interaction, even if it is just sitting quietly in the room on your phone. After 6 months you might try just removing the gate and see what happens, it might give her more incentive to come down. Hissing, growling and even some tussles are normal, just not bites and deep scratches. I wish you well.....