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Yes, you can DIY either core or liquid aeration for your lawn. You can use a manual core aerator or rent an automatic one. Just be sure to mark off yard obstacles like sprinkler heads before you start. Liquid aeration is easier—just hook up a garden pump sprayer, although some are able to be attached to a hose. While liquid aeration is not as laborious as core aeration, you will need to focus on an even distribution of the solution across your entire lawn for it to be effective. Of course, if you are unsure about doing both, call a pro.

Post-seeding care involves regular watering, mowing when appropriate, and occasional re-seeding to cover sparse areas.

Whether or not grass will fill in bare spots itself depends on the type of grass you have. For example, grass with rhizomes (e.g., Bermuda grass) and stolons (e.g., Saint Augustine grass) will naturally fill in bald spots as it spreads laterally. However, clump-forming grasses will not spread. 

You can try that, but it isn't likely to yield results. It's like watering potted plants without putting the seeds in. Most of the tossed grass seeds sit on top of the soil or existing grass blades and never germinate. The seeds need to make it into the soil, just like growing a potted plant. The best way to accomplish this is to aerate your lawn first, then overseed.