I can't imagine even doing the new mow lines unless the tree removal is done. It's critical to establish sight lines, and that doesn't happen with trees in the way. Also, from an agronomic standpoint, I wouldn't reccommend cutting anything shorter unless the shade and other tree nasties (roots, competition, etc.) are not an issue.
Start with tree removal, and make it wholesale clearing if the budget allows. Then, make your next move with mow lines, green expansion or renovation, then bunker work.....and perhaps decide whether new tees are wanted.
Joe