I'm sorry you're so unhappy, PRJC. It's hard to advise because only you know what's going on here... But I would say this...
Be careful about scapegoating your step-daughter for the problems in your relationship with her mum. For me, big alarm bells ring when you say "On top of this her mum and I have not had any sexual relations for about 18 months". This can't possibly be the fault or responsibility of a twelve year old; it's for the two adults to sort out - or not.
What you describe is an insecure child who's jealous of the attention her mum gives you, and who is terrified it means she's going to lose her mum, so is fighting to keep her attention. That's natural. And it's not her fault that her tactics are working. In fact, it will probably be making her feel more insecure and afraid, because when her mum 'gives in' to her manipulation, she is (unintentionally) giving the girl the message "mum feels guilty so girl must be right" ... To feel safe and secure, she really needs you and her mum to work together to show her there is no competition between her and you, and no chance at all, ever, of you 'stealing' her mum from her.
Teenagers are hard work. If she's only 12, the chances are, her behaviour is going to get worse before it gets better. You and your partner need to work out how to deal with any bad behaviour together - or maybe rather, work out whether you actually can - and not blame her if you can't.
It's difficult being a step-parent. Sometimes I feel I have only continued to 'stand by' my challenging teenage son because he's my son. If you don't have that 'blood loyalty' I don't know how you manage. Good luck.