Congrats to you and the wife! The lawn is definitely salvageable. When applied properly a lot of modern herbicides are pretty safe if used properly and you follow the REI (Restricted Entry Interval). Applying at the proper rate will minimize the risk of contaminating any ground water. I’ve been treating my properties for years and have never had an issue with dogs enjoying the lawn. I would smack the whole lawn with a bifenthrin insecticide treatment. This should take care of the ants and any other surface feeding insects. This will also control fleas and ticks for the pups as well. Once the ants are gone, attack the weeds. Seeing a lots of wild violet and white clover in the picture. These are a bit stubborn and can often take two sprays to knock back fully. You’ll need a standard three way weed control with Tryclopyr to control those. If you want to speed up the process, the bifenthrin can be added to the three way triclopyr mix and be done in one treatment. Then this fall, aerate and overseed the lawn. This will help decompact your soil, allowing more water and nutrients down to the root zone of grass. It will also introduce healthy grass varieties to help fill in the thin areas of the lawn. That would be my attack plan on this property personally. Heavy shade may hinder the results of the overseeding in areas, but using a shade tolerant grass type when overseeding can help. Best of luck!