I am a stepmother and I can relate.
The problem here is with your husband. He doesn't want to deal with the hard part of parenting a teenager and/or he feels guilty about his daughter having 2 homes. If he's really taking her side, that's a problem. He needs to support you, his wife, for several reasons - the obvious ones, but also that his daughter needs to see that all women deserve respect from the men in their lives so that she doesn't get involved with some guy who doesn't value her or her opinions. If your husband disrespects you, she'll learn that it's okay for a man to do so.
If you were understandably short-tempered at one time, you can own up to that in counseling and admit that you weren't the warmest person around - you were nervous about a new baby, exhausted, learning your way, etc. Maybe you gave your stepdaughter a bit of short shrift. That said, it doesn't make all the problems your fault.
I think you won't get anywhere blaming the mother for not taking care of EVERY holiday and the summer. If she works from home, she WORKS. I do too. I couldn't do my work if I had a child around 24/7, much as I love my child. So don't say "NEVER" about having your child in after school care - even if you are sure you wouldn't do it (and who knows what life will be like for you in 3 or 5 or 8 years?), you don't get anywhere criticizing another woman and her choices. Do not try to make this girl feel badly about her mother - you don't like it when you are criticized and undermined, so don't join in that trap. It is disrespectful, and it will backfire.
If the daughter is complaining to the mom on the phone so that you can hear it, understand that is PLANNED to make you feel badly and/or guilty. Don't buy into it.
If you stepdaughter is manipulating your husband, I say again - HE is your #1 problem and you need to develop a strong relationship there in which he appreciates you and in which he takes full responsibility for his daughter. You and he need to agree on what is safe for your little one, and present a united front to the stepdaughter on what the rules are.
You can ignore the snotty behavior or say that you won't stay in a room where you are disrespected BUT she may continue to do it just to force you to leave. The thing to do is not take her to the mall or do her laundry or any other things she wants until she straightens up. But again, you cannot do this effectively without your husband's involvement and support.