The answers to some of your questions are somewhat subjective, so you might get some conflicting answers to them. #1. There are a couple of ways to go. In my opinion, the best way is to remove the sod, fill, and re sod. You could do the more or less the same by reseeding, but growing new lawn from seed while cheaper, is a PITA as far as I'm concerned. Personal opinion. #2 Lawn care companies in my area use a special fertilizer in the late fall to get a jump start on green-up in the spring. What to use really depends on the soil in your area. I would discuss this with a lawn care company and perhaps have the apply it. Aeration is better in the spring, although it would not hurt to do it at anytime. Grass should be mowed fairly short in the fall to help prevent snow mold. This may depend on the species of grass you have. I mulch my grass so thatching is not necessary. I believe if you do it, it would be better in the spring. I strongly encourage mulching. You do need a mower that is also a mulcher as mulching is not the same as just leaving the clippings on the ground. #3 By now weeds have pretty much quit growing which means chemical weed control basically a waste of time. It's too late for crab grass control also. The best weed control is growing a healthy lawn and not cutting ti too close (except late fall, as mentioned) The taller thick grass shades out weeds. When you mow just even the tops. This also conserves water because the shade slows evaporation. #4 Mushrooms are a real problems that I know of no cure for. Other fungus may have a control, but again a lawn care company could provide better advice for your specific problem.
I have found it worth while for me to have a lawn company handle the fertilizer and weed control in my lawn. It does cost to do that, but they have the right type of chemicals, some of which you have to be licensed to buy and use) they know when to apply it, and they have the equipment to apply it. I do deal with noxious weeds around the edges of the lawn with Round-Up myself, and I control weeds in my rose bed, and other shrubs with Casoron, and I fertilize my roses and and other shrubs myself.