I see three possibilities: lack of social education (probably passed along from previous generations); a princess syndrome (fostered by overindulgent parenting); or this girl (and possibly her mom) are on the autism spectrum (Aspergers types, who are frequently oblivious to social cues from other people, and can be VERY particular about how things should be done).
The daughter is not at fault for any of these if she's younger than, oh, say, adult. Even then, childhood habits and expectations are hard to recognize and overcome. (How many of our own attitudes and habits have we successfully changed?)
So don't take it personally. She's not out to make your life miserable, she's only trying to meet her own needs (as she understands them). Just return the favor, and be clear about your own needs when you're interacting with her. You don't even have to tell her she's being rude, because if she's able, she'll gradually get it.
You can give her messages like, "A is playing right now. I'll ask her later whether she'd like to visit you." (Or, if it works for you, you could invite M over to join the other friends.) or, "I'll ask your mom if you can sleep over – here's what I'm serving for dinner tonight. … Oh? No.If you want to stay here, that's what you'll be eating for dinner." or, "I hear that you think pancakes should be made your way. At my house, I make them my way. You can stay here and enjoy pancakes with us, or you may go home for breakfast. Should I call your mom and let her know you're coming home?"
(This last issue sounds particularly like the child is an Asperger's type.)
All these things can be said gently, in a friendly tone of voice. Anger from you might be puzzling, because I doubt this child knows she's doing anything wrong. And by using calm and polite responses to her demands, you'll be modeling good manners yourself.
Talking to the mom isn't likely to make much difference, other than creating an unnecessary rift. Which could be the fodder for neighborhood gossip, division, and drama. Who needs that?