Ah, now that you're using £ as your currency in your posts, I can tell that you're in the UK which is wonderful news! It always helps to mention your location because there's an automatic assumption here that you'll be in the US, and a lot of the advice is US-centric.
See, your local council is responsible for housing you. They can provide emergency housing to get you out of an abusive situation, and they can provide long-term housing so you'll have your own place to stay.
So all you need to do is to contact your Council's housing team, explain you're in an abusive situation with a step-father who's threatening to beat you up & to commit suicide. Tell them you're frightened for your own safety & that you feel you're in immediate danger. Also, tell them about the dogs and ask whether they have anywhere where you could take the dogs with you (you do call them 'your' dogs: are they yours, or do they actually belong to your step-father?). They'll arrange emergency accommodation for you & will get you on the housing list so you can have a place of your own.
Then go to your local Citizens Advice Bureau to get advice on other accommodation options; benefits; domestic violence assistance etc. Yes, you are currently being subjected to domestic violence: emotional abuse and blackmail, and threatening suicide if you leave, are all forms of domestic violence.
Then as soon as you find a new place, just walk out with your stuff without saying a word to your step-father. Block his numbers & social media so he can't try to blackmail you into returning. Tell the rest of your family that you've moved out because you can't stand his abuse any more, and tell them you don't want him to know where you are and you don't want any contact with him. It also sounds pretty ominous when you say you have reasons to know why he's bisexual & that he doesn't want you to look feminine. That suggests he's either sexually abusing you, or that he wants to.
He's not likely to change, and that counselling he's supposed to be going for in April is not likely to make things better for you. Don't hold on to false hope: please, get out as soon as humanly possible. Then if by some miracle he does change, you can always visit him on occasion to keep your relationship going. But never move back in with him, because it'll get worse every time you do!