Not familiar with Texas Wesleyan, but I would advise avoiding Texas Southern. http://www.ilrg.com/rankings/law/view.php/96
The school has had some serious trouble recently (accreditation, etc.). It also has the worst bar passage rate of almost any law school. So for that school specifically, my answers would be:
1. If they do, likely unpaid.
2. Very, very, few recruit there on a regular basis.
3. Yes, try South Texas if you want a 4th tier school in Houston
4. Unless you are one of the top 5 (not top 5%, top 5 people) it's unlikely anybody will recruit you from Texas Southern. You would most likely need to pass the bar and hang a shingle.
Not trying to hate on anybody. I'm sure the school has and will continue to produce some great lawyers. Just fair warning.