Much too soon to be picking specific medical schools. You won’t know for a few years whether you’re competitive for BCM.
RE: retaking college algebra. Unless you did poorly in pre-calc and feel you have a weak math foundation, you should go ahead and take calc. Taking college algebra (which is basically high school algebra all over again…) could be construed as trying to pad your GPA and med school adcomms don’t look favorably on that. Also since UH requires its bio majors to take 5 (http://catalog.uh.edu/preview_program.php?catoid=6&poid=1578&returnto=1129) math classes, taking college algebra will just cut into your available electives and delay your required math sequence completion.
You can contact profs about research positions now if you want, but don’t expect to get many (or any) responses. (It’s summer! Even professors get to take vacations.) If you learned any specific lab techniques/skills at Rice, be sure to include these on your CV.
Consider that you may want to hold off on getting involved with lab research and taking on additional activities until you see how well you’re adjusting to the college workload. You don’t want to crash & burn and be one of those posters who comes back in year asking how to fix a poor freshman GPA.