You might kill the tree, plus any bees and butterflies in the area which would certainly result in less fruit. I would take a piece of landscape fabric around the tree and put some mulch on top.
I wouldn't. Pull out the grass, put down landscape fabric and mulch.
Hello my name is Peggy. Yes you can stray round up under your tree, but you will kill grass and weeds. Try not to get it on the tree leaves or trunk.