Briefly, you fill in bare spots via overseeding. Simply putting down more seed in the bare areas. The critical part is making sure the seed is firmly in contact with the soil. A few ways to do this are slit seeding which uses a machine to make a 'slit' plant the seed and cover it. A second way is to core aerate. This is another machine which removes cores of soil. This is usually done to reduce compaction of soil, but with those open holes you can overseed and the seed washes into the holes with the first rain and the plugs of soil left on top of the soil erode and cover the seed.

The most common way is to use a garden rake to slightly rough up the area, spread seed and then press it down. You can buy a roller that you fill with water and roll over the area or rent a metal roller that you can push/pull or attach to a riding mower if you have one. Otherwise just tamp it down on moist ground with your feet.

From there keep the top layer of soil moist until the seed germinates.

If your grass is dying due to excessive water though, there really isn't much you can do about that. It sounds like you have already addressed the water issue as best you can.

The tree can be trimmed anytime. Usually new grass is considered 'established' once it has been mowed twice. Not really true, but it's the practical rule of thumb.

Please do head over to the lawn care forum though as you will get a lot more and probably better suggestions.