Here's a link to a listing, courtesy of the B'ham Public Libraries, of various shelters for abused women and children, as well as other social service agencies:
Birmingham Public Library - Virtual Library - Subject Resources
If you're in a truly abusive relationship, you need to get yourself and your children out immediately, and then deal with finding a job and apartment. The folks who work at the places linked above can help you in your job and apartment search, too.
If you check the website for the local newspapers, al.com, you'll find a job listing link where you can search for jobs and even apply online. You'll also find a search page for rentals. I can't answer your questions about pay or rent, but if you check out the al.com website you'll find those answers. Once you get to B'ham, you can pick up apartment guides at any convenience store and most grocery stores. If you make $8.hr., I'd guess that you'd be looking for an apartment that rents for no more than $400/mo., and preferably more like $350. You may want to look into govt. assistance programs for low income families.
The most important thing is to make sure that you and your children are safe. Then you can deal with the rest. Best wishes.