Oh, you poor love.
Some stuff you can do, some you can't.
You can take your child to live anywhere within England and Wales if you're here already.
You can't change your child's name without your ex's consent because he has parental responsibility. You need him to sign a letter agreeing to the change of name, and if he won't then you could make an application to the court but to be honest it probably isn't worth the bother at this stage.
You will be entitled to some benefits and I suggest that you go on the website EntitledTo.com.
So far as the separation agreement is concerned, it may be overturned by the court as invalid if one or both of you didn't take legal advice and didn't provide formal documentation of your assets, however as you're saying that you have no assets to divide, I'm assuming that the sep. agreement only covered care of your baby, and child maintenance? Essentially, neither of you are really going to be bound by a document you've drafted between you without legal advice. This may or may not be a good thing depending on what it covers. If it's just child care stuff, tbh, I wouldn't fret too much about whether it's binding, firstly because a lot is likley to change over the next 18 years and secondly because the court always has jurisdiction to make whatever order is in the child's best interests if there's a dispute, and the CSA is there for the money side if he doesn't pay his maintenance.
If your ex isn't paying regular child maintenance of 15% of his net income then you should contact the CSA.
When you're ready to think about divorce, you should firstly go to the website www.resolution.org.uk to find a local family lawyer. Tell them you want a free half hour interview. You will almost certainly be entitled to public funding if you have no assets and no income, so make sure that the firm you're going to does public funding.
Good luck.