While I agree that these would be seen by most people as parenting differences... this is not a cut and dry situation. This is a father with a serious psychiatric diagnosis, who is unmedicated, and a child who is also having psychiatric issues.
Also... whether to allow many of the things Mom listed are dependent upon the child and the child's state of mind/maturity. For example:
Twain is my 8 year old boy. He'll be 9 in January. SK8 is my eight year old stepdaughter. She will be 9 in December. Twain is very independent, has a great sense direction, is very responsible, and has a brain like a homing beacon. He is allowed to ride his bike 1/2 mile or so to the park, walk to Sunday school, run into the grocery store with a short list and some cash, etc. On the other hand, SK8, has no sense of direction, and is distracted easily. I'm not kidding when I say she got lost walking to the end of the block to attend a birthday party. I'd *never send her into a large retail store and just let her wander. I might never see her again, and not because she got snatched, just because she wandered off.
It doesn't sound to me like this child is quite ready to be left to her own devices, especially in crowded overwhelming places.