I'm sorry you're in such an upsetting situation. She sounds like a typical teenage girl...except she's getting away with more because you're not her biological mother and her father feels like she should get to choose where she lives.
If you and he were both her bio. parents, you could insist that she continue calling you Mom, because it's rude to call your parents by their first names; and your DH would probably tell her moving in with Grandma is not an option, that you guys are her parents and she will have to live by your rules until she's ready to support herself.
When teenagers test the boundaries like she is, of course they think they want freedom. But (my theory is that) they feel more secure and loved if their parents stand up to them and enforce the boundaries.
Think about toddlers. They push boundaries and get outraged when parents reinforce them, because they want to do what they want, when they want. Yet, it's obvious to parents AND to the toddlers themselves that the toddler's world would be pretty scary if they ACTUALLY got to be in charge of everything. So, even if only subconsciously, it's reassuring to toddlers to see: "Even when I act completely rotten and make it MISERABLE for Mommy and Daddy to enforce their rules, they STILL won't let me do things that are bad for me. Because they LOVE me. So it's SAFE for me to be a kid and test new things b/c I can TRUST my parents to stay in charge, even when I make it hard."
It's SOMEwhat different with teens, because they DO need to be taking charge of some things in their lives. But, deep down, they know they're not yet ready to be COMPLETELY in charge in their own lives. So it's still reassuring to know their parents ARE still in charge; that they can trust their parents to look out for them and OVERRULE them, when they're doing the wrong thing - EVEN if the teen makes it absolutely miserable for her parents to take care of her!
I also think when a teenage girl stops calling someone "Mom" and starts using her 1st name, she is testing whether the relationship can now be one of equals. I think it would be comforting for her to have it reinforced that, no, the relationship hasn't changed. You are still her Mom. You still have protection, guidance and reassurance to offer her that is over and above what her equals - her friends - can give her. You still have a love for her that's so powerful and enduring that you deserve a special name that she doesn't call anyone else. Even if she's a PITA, you are not abdicating the role of mother. You're in it, thick or thin. However, it IS awkward for you to demand that she go back to calling you Mom, when you both know you're not her biological mother.
HOWEVER, your DH could definitely make her move home! The two of you could sit down with her and explain you made a mistake. Until she's old enough to support herself, she's not old enough to decide where she's going to live, or whether she's going to have boys in her room, or how much she's going to party. You love her and, in setting rules that seem inconvenient or uncool now, you are thinking about her future: trying to prevent early pregnancy, or partying so much it affects her school performance. You don't want the quality of her adult life to be hampered by her silly decision-making and priorities, at 15. It doesn't matter if she agrees with you, or likes it! You love her enough to LET her be mad at you. You understand that it's normal, at her stage of life, for her to try to break rules and get made when she ets caught. You don't think she's bad. You won't stop loving her. But she needs to understand it's YOUR role in life, to give her good boundaries, until she's mature enough to take care of herself. So you hope, somewhere deep inside, she knows it's from love and not just a desire to annoy her!
