Per growing Zone's, you are probably in a transtion Zone 7-7A, you might even have a sub area to 6. It depends on what is around you. Lots of trees? lots of buildings/homes/highrises? can even change your temps.
Fall is the best time to seed, it gives them time to germanate, put roots down, and get ready slowly for summer. Now if you get snow, and if so, how soon will depend on if you should plant this far into the fall season? If you are able to wait on planting or seeding right now, I would. But if you live in an area where you have an association that governs how your neighbor is kept, they may have a time frame that you have to follow on only having a dry lot of so long.
If you are able to wait, and due to the clay soil. I would spend the remaining fall season and winter, reconditioning and amending your lawn and flower beds. When you get your compost, steer manure, peatmoss (is a must). sherdded leaves, sand, and on a time when the lawn areas are dry and you have a covering of 3 or more inches of the stuff, I'd rent a rototiller and go to town. You can get to a depth of 5-6 inches deep.
Ok, 3 inches is a lot and can be expensive, but check out the mom and pop garden centers, your yellow pages, or stop gardners. Ask for the name of or a place that sells bulk landscaping soils. (your best buy). If not just get what you can in bags, put it aside, get more, then put down.
Now is the time to put all your effort into inproving the ground, before seed is put down.
If you chose to let this company do the core aeration. They should core every 4-6 inch apart to a depth of 3 inches, more is better. Then I would apply your mixture of compost/sand/ peatmoss on top and water.
Remember, companys are out to make money, pay their bills, put food on their tables and employees tables. Yes they do have to have good reps to say in business. Especially now in this time of hurt. But when they tell you, you will probably have to change out to another time of grass if you quit them.....I'd think twice, if not more on using them on suppling you your grass seed.
If you have the time and patience, and the determanation. Heck your getting compost! Most people don't even know thats one of the most important items you need for a healthy base on having a lawn. (Also the more organic compost you use, the better) Don't pinch on this. Compost will save you money in the long run. Clay is a heavy base soil and when watered. Seeds get boged and rot. Compost and sand will fluff your land. Along with feed.
Get your soil tested......find where your Agriculture extension office is. Take samples of your soil to them, before you add compost. It's the best way to find out what types nutrients, pH soil type you have. Then 6 months later do it again, to see what is needed. The Ag. extension also can even give you heads up on what type of grass variety or blend of grass is best in your area.
Can your aeration be done by the do-it-yourselfer? Oh, yea. Aeration can be done by any home gardener. Rental stores, and some major home centers rent gas powered walk begind aerators. But a lot of homeowners do have someone come and do this one thing for their lawns.
All in all, I'd check in with your local code enforcement to see what your size requirements on having lawns are. Have the mininum, make larger flower beds. Take my time on adding admendments NOW with a rototiller, or if you can't wait have the company come and core, you add admendments. Get the grass the Ag ext. Office suggest. Put down and top dress with a mulch to keep seeds moist and safe from birds.
Also ask local nurserys pick their brains, ask for ideas. JUST ASK. oh...and that neighbor down the street with that beautiful yard... is now your best friend.....ASK, ASK,ASK...HOW, WHAT AND WHY ???????
Hope this helps, and Happy Gardening.