Hi SR,
It's a matter of personal choice.* If you don't like the look or need to walk under it, then you can remove the lower limbs.* Personally, I prefer them on, escpecially since not much grows underneath anyway the the trees roots are near the surface.* Here's some info on your tree.
http://hort.ifas.ufl.edu/trees/PICABIA.pdf
If you need sites on how to properly prune and mulch, just let me know.
Newt
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.