Ok and thanks for the extra info.
I can now reassure you that he cannot stop you moving house and taking your daughter with you.You have the civil liberty to live where ever you want in the UK. As you have been your daughter's main carer for the last 4 years, your ex would not suceed in getting a court order to say that your daughter should now live with him - unless your daughter was not safe in your care and/or not being looked after properly by you.
In theory, your ex could apply for a court order (called a specific issue order) to say that your daughter must stay at the school she is currently at - if he does make such an application, your response to the court would be to explain why you need to move, how the new school will be at least as good as her current school in meeting your daughter's educational needs, and that she will continue to have good contact with her father.
Mothers automatically have parental responsibilty. Fathers can acquire parental responsibility in various ways - one of which is by being named as father on the birth certificate. Having parental responsibilty is largely symbolic - but it does mean that you are obliged to consult the dad about major decisons in your child's life - eg chnaging her name, emigrating, or changing her school. If her father does not agree, then you do need permission from the court to go ahead without his agreement. But I think that the court would grant permission as long as the new school is a decent school, and you have good reasons for moving, especially as you are not moving very far.
With regard to your proposed reduction in his contact, it is government policy that children should have contact with the parent that they do not live with - unless there is very good reason why not - but the precise contact arrangement depends on the particular circumstances in each case. Parents are supposed to reach a joinnt decsion about the contact accroding to what is appropriate for that child, and workable for the parents. If the parents cannot agree, then either one can apply to court for a child arrangements order. Given that you have given him the option to continue the contact as it is, as long as he comes to collect your daughter, but he has refused, I think the court would be sympathetic to the change you propose.
However, going to court is stressful, time-consuming and expensive, so if you can negotiate something instead either between yourselves or via solicitors' correspondence or via mediation, then that is preferable. The family court anyway now requires the parties to have attempted mediation before it will consider an application to court. Here's where to find a local family mediation service:
You would also benefit from some face-to-face legal advice. Here's where to find a specialist family law solicitor:
Whether or not he pays a penny towards your daughter's needs has no effect on how the court decides all of these issues.
I hope this helps and I wish you the best of luck.
Thanks and best wishes...