You have my sympathy, you're in a daunting situation. Fortunately (or not) once you start putting the wheels in motion to leave, the next essential steps tend to show themselves.
When I split from my husband, I was unemployed and living in his home. You can claim Universal Credit as a single person once you've split up, even if you still live together.
Once you submit the claim, you will be invited into the Job Centre for an interview with an assigned work coach. They will talk with you about your circumstances. Since you are self-employed, they will need to see evidence of earnings.
Bear in mind, if your pay is under the table or the situation of your self-employment isn't fully legit, your ex-partner and his mum might get very agitated to know that you've submitted a claim, as the business will be put under scrutiny. Still, you need to put in your claim. If you earn under the earnings threshold (whether during or after the self-employment) you wll get some assistance with housing costs, and this will help you to afford private rent.
I would also prepare a bug-out bag, just in case: important documents for you and your daughter, overnight necessities. In case you need to leave quickly. My separation began very amicably, but when the reality of the situation hit my ex, he became very unstable very quickly, and I had to do a flit for my own safety.
Let a trusted friend or relative know that you are leaving your ex, that they can host you and your daughter overnight or longer, should that be necessary.
When I did my flit, on the way to my aunt's, I reported as homeless to the Council. I was already on the housing list as a Band C because prior to leaving I was "adequately housed" - I'm sorry, you're never going to reach the top of the list as a Band C. And I expected the Council to find me "intentionally homeless" (so in hindsight, I can't really remember why I went to see them, beyond a vague sense that I "should"). But, when I described the situation, the Council identified my situation as having elements of Domestic Abuse - I was allocated a caseworker, who helped me with benefits application forms, and told me about a referral-only Council scheme to place working people on the housing list with private landlords (I was in my first month with my new job by then).
My point is, there is help and support and guidance out there.
I am ADHD and autistic. When my thoughts are swirling, I find it so helpful to empty them onto a page. Here I identify what's a priority, what isn't, and where I'm lacking knowledge.
Try not to worry about your daughter's schooling, not yet anyway. It's a big, important topic, but it's not your immediate concern. She will be okay, even if you have to move schools.
Don't be afraid to tell your daughter's school that you and her father/step-father are separating. They will be prepared to support her emotionally, and that will give you some comfort.
Even if you move to a less desirable place, there is a real joy to be had in having and making a home that is just yours. I was once a homeowner, now I rent privately. I should be devastated, but I'm not.
I have kids too. We have 50:50 custody. The kids are fine (their dad's poor behaviour towards me hasn't extended to them).
So... Get your paperwork and finances together. Have a safe, temporary space to go to if necessary. Give yourself breaks to gather your thoughts (coffee and cake after each big appointment!) Speak with people, friends and professionals, because even if the separation happens as smooth as silk, it will still help to know that people have your back.
You can do this!