1. Would removing the above ground segments hurt the tree significantly?
2. Any easy method of removal other than getting a stump grinder after
them?

1. I have 13 big old oaks on my property and have taken out a few
roots to put in gardens, remove hazards, whatever, with no affect on
the trees. The roots I removed were only an inch or so in diameter
that I clipped with a lopper. I think the key is not to take too many
thick *major* roots from any one tree.

2. I dig away the dirt and use an axe for the thicker roots. Do you
need to go really deep to run pipe or just to replace the root with
some soil and turf? A stump grinder's a pretty serious piece.