I don't want to say the company is planning to rip you off, but it does seem like they are offering to do a lot of things that are not that normal. Anyone can scatter seed. Power seeding, if it's what I think it is, is only done on huge areas or hard to reach spots on a hill. At the end of doing all that, you will still have a lawn full of weeds. And most of what they quoted you is not needed.

Normal lawn care consists of
1. watering a full inch once a week in the summer heat transitioning to once per month in the cool months.
2. mulch mowing once per week when the grass is growing (I would suggest mowing at your mower's highest setting unless you have a dwarf variety of turf grass).
3. Fertilizing 3x per year with chemical fertilizers or (?x per year with organic fertilizers)

That's all...for the most part. Topdressing is not needed unless you need more weeds in the yard or you need to redirect the drainage back into your basement. Core aeration is not needed unless you feel the need to spend $228 instead of $0.50 (see below). Weed n Feed is not needed, ever, under any circumstances.

However, if you have a fescue lawn and it becomes thin in spots, then you will need to reseed in the fall. And that might be every fall until the grass density is enough to keep the weeds out.

If your soil is hard right after you water, then you might want to treat it with a surfactant to allow better water penetration. This treatment is taking the place of expensive core aeration. You can get expensive surfactant for $75 per gallon at specialty lawn companies or you can spray any clear shampoo at a rate of 3 ounces per 1,000 square feet. Those of us who have used the shampoo are the biggest proponents of it. You need to spray the shampoo twice, 2 weeks apart. Do it right before you water. This treatment should last for years. You'll know it is working when your soil feels soft for several days after you irrigate.

You might need lime but you also might need many other micro nutrients. The ONLY way to know is to send a soil sample to Logan Labs (not Penn State). Unless you are a Penn State grad and feel the need to keep them in business, please at least look at what you get for your money at both Penn State and Logan Labs. The $20 test at LL is much better than any of the state school tests we've seen. The only place where Logan Labs falls down is in interpreting the results with suggestions for what to do next. Fortunately there are Internet lawn forums where they have read hundreds (thousands?) of Logan Labs reports and can tell you exactly what you need to get on eBay/Amazon to treat YOUR lawn. Search Google for "logan labs soil test soil management"

So here's what I would do if I was doing it myself.
1. If you have not fertilized since the spring, fertilize now at half the bag rate (for chemical fertilizer) or at 20 pounds per 1,000 square feet (alfalfa pellets - an excellent organic fertilizer).
2. In two weeks spot spray the broadleaf weeds with RoundUp. Any other time I would suggest using Weed B Gone but there is a time delay on reseeding after spraying that stuff. With Round Up you can get going faster.
3. Rake away the dead grass and weeds and mow low.
4. Scatter the seed in the dead spots and anywhere the grass is thin.
5. Walk on every inch of the new seed to push it down into the soil.
6. Water it 3x per day for 3 weeks if you have only fescue and 4 weeks if you have a mix with Kentucky bluegrass in it. That should give you 80% germination for the new grass.
7. Set your mower to the highest setting and mow when it needs it.

In a few weeks you can tell if you missed some spots. By starting now you will still have time to repeat the seeding before winter sets in. NOW IS THE TIME TO SEED COOL SEASON GRASSES. Do not wait for spring. Spring is the time the summer weeds are germinating.

When to fertilize the new grass???
I'll make it super easy for you. If you use organic fertilizer (alfalfa pellets) you can fertilize any day (or every day) of the year. If you use chemicals it's practically a crapshoot. Alfalfa will not burn anything, so you can't make a mistake. When Thanksgiving rolls around, you can fertilize with a high N, fast release, chemical fertilizer as a winterizer. If you really feel the need to do something in the early spring, use alfalfa pellets again. If you can get through the early spring with doing nothing, then apply the fertilizer of your choice on Memorial Day. Generally lawns do not need any fertilizer in early spring. That's why you wait for Memorial Day. If you feel the need for a preemergent in the spring, write back and we can discuss it then.

A note about seed.
You get what you pay for with seed. Excellent seed has 0.00% weed seed and 0.00% Other Crop. You can only get that on the Internet. If you want a showcase lawn, then do that. If you just want mostly green, you can get what they have at the box stores and deal with weeds later. It's not that hard to deal with weeds if you are doing the watering and mowing right.