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.
If you need sites on how to properly prune and mulch, just let me know.
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.