Many of those "weeds" are also wild flowers. The most environmentally friendly, organic, method of "weed" suppression is to deprive them of access to the sunlight they need to grow, cover them with a good thick mulch, or newspaper, cardboard, etc. and cover that with something, vegetative so it also contributes to the health of the soil, to hold the paper in place and hide it. Tilling will quickly mix the present growth with the soil but will also turn up many "weed" seeds that will compete with what you plant there. Some people will till now and then again when new growth appears and then again when new growth appears until no new growth appears. There is no environmentally friendly spray material that kills plants.

Yes, I do see the dandelion leaf in that shot now, I think. And the small dandelion plant in the last one. They very well could be multiple things there. I will look at some Erigeron species and compare them to what I have outside. Thank you very much, again!

I use a machete or hatchet on a tree stump or old wooden cutting board to reduce vines and tough stalks to pieces I can turn with a pitch fork or broadfork. A sharp spade works too, on ground you dont mind chopping up. I also find its easier to chop as I go, snipping vines with the nippers as they go into the bucket rather than piling vines and having a tangle of them. The nippers get sharpened more often but sharpening is easier work than turning the compost or chopping. Salad bite sized leaves and twigs will break down very quickly. I like one hot compost near the kitchen yard for kitchen scraps and weed seeds. Its worth the chopping. Even in the larger compost, large leaves and stalks can block air and water and slow the process. Its pretty hard on the mower to use it as a chopper.

I use Milorganite in the spring as well, for summer and fall I use the Scott's Turfbuilder product designed for that season. I am by no means a lawn expert though, these are just easy to find, relatively inexpensive products that do the job well enough for me. I'm confident you could kill the ivy (and not your trees) if you have the time and patience. Again I'd suggest trying the Weed B Gon Chickweed Clover & Oxalis concentrate in an Ortho Dial N' Spray hose end sprayer. Just pick a calm day and get down low to avoid spraying the tree trunks and leaves. If the ivy is anything like wild violet it will take a couple of applications but it will work eventually. Then like you mentioned you'll have to rake out the dead ivy and reseed the bare patches. But as others have said, a weed-free (or even weed sparse) lawn is a huge commitment so I don't blame you for hiring it out--if you can find someone to call you back. Last thing I'll say is that no matter how diligent you are about your own lawn, if your neighbors aren't equally diligent you will always have some of their dandelions, clover, ivy, etc. filtering into your lawn. So I've lowered my expectations and live with a semi-weed free lawn after 13+ years at my current house.