I think the grass will be really rough on the trees and I'd definitely
mulch. It
should make a big difference in all sorts of ways.
I'll add another question to Jeanine's...can you find out/do you know if if it's a 'running' or 'tufting' grass? The runners have extremely aggressive mats of roots, generally shallow and horizontal.
If it's tufting, I find it extremely easy to kill just by smothering it with plenty of mulch. Running's a
lot harder, and I'd lay thick, seriously overlapping
cardboard. A running grass
lawn will
always be working its way back into the mulch, so prepare to be vigilant!
Whatever grass it is, I wouldn't dig it for many reasons, not least because if it's tufting, you don't need to, and if it's running, you'll just encourage it
Trees like a fungally-dominated environment; tree trimming mulch is an ideal high
carbon (hence fungus-friendly) mulch.
A caveat: Be careful to leave an area clear of mulch around the trunks to avoid collar-rot. I haven't had any problems butting card right up against trees to kill grass, but maybe I've been lucky.
I'd add a few support species too, like comfrey, alliums, spring bulbs, clover...