looks like an oak to me, but if you want to know download an app called PictureThis, go take a picture of the leaves and it will tell you what tree you've got
i think if you cut the limbs high enough for you to walk under the tree you would not hurt it
Make the first cut up from the bottom several inches away from the trunk, make the next cut down from the top of the branch to meet the first cut, then make the final cut down from the top, cutting just outside the branch's "collar" - the swollen part at the base of the branch.