don't till the soil under the drip lines of any trees. you can and will do tons of damage to the roots of the trees. aeration with over seeding is what you want to do. and as far as the spots directly under trees that get no sun and would need endless water i would do a mulch bed.
also the tree in the 5th picture down i would get that cut down asap. hate to tell you that but if that was on my property that would of been removed before the papers were even signed, that tree scares me that much.
can't tell you what kind of seed to use not sure of your zone and if its cool, warm, or another grass. i would say call lesco or another fert. distributor in your area and see what type of seed the recommend.