Answer: I think the first approach is to eliminate the weeds and the ant hill, then rake the bare areas and reseed the entire lawn. I understand your concern about chemicals and pets. All of my research indicates that glyphosate (Round-Up) and Scythe (pelargonic acid and related fatty acids) are safe to use on lawns as long as you keep the animals off the grass for a day or two after application. Round Up will kill dandelions all the way down to the roots so it is the most effective product. Others will kill the tops, but the roots can send up new shoots a week or two later. So, spot treat the broadleaf weeds with Round Up (it will kill any vegetation it touches so don't spray it all over the lawn, just target the pesky weeds).
After getting rid of the weeds you can aerate your lawn (rent an aerating machine at your local garden center). If you have the money and are so inclined you can spread some sand over the lawn. The plugs that are removed during aeration will decompose and any sand you spread will work its way into the holes the plugs left. This will help your lawn drain well and loosen the soil enough for the roots of the grass to easily penetrate. Water your lawn regularly and the grass should green up and become thicker; reseed the bare areas and it will fill in quickly.
Best wishes with your lawn!
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