First things first, the cage is too small for a grey, you need something bigger and not to be placed on top of a cupboard, she needs to be down lower.
A week is not long enough for her to get used to you and being in a new home, she needs more time to settle in and having those boisterous dogs around all the time won't help things much. Can you pen the dogs up for a while every day so you can have her out of the cage, she needs some one on one attention from you.
Why did you get a grey for if you are afraid she will bite you, that is just part of grey ownership, we get bitten from time to time but she needs her food and water replenished every day or you will starve her to death.
She is growling because she is scared and new to her environment, give her time to calm down and continue to talk softly to her and offer her treats. She won't feel completely safe with you until she learns to trust you and you have to earn that.
She is still a baby at 11 months of age and it might take a little longer but I think you can work out a reasonable arrangement if you put your all into it.