Technically, seed germination no, establishment yes.

All a seed needs to germinate is water but once it has germinated and needs to get nutrients from the soil proper pH will make the nutrients in the soil more available to the grass plant.

Did they tell you what your soil pH is? Less than 6.0 pH will have an impact in seeding.

That said, lime doesn't work overnight. It starts working right away but full results usually take months. You might want to try one of the fast acting limes. If you google "fast acting lime" you can find more information.

Cat, its too late now for considering seeding a lawn....the time to do that was back in September....
Its not a case of whether the weather is good, or bad, its whether the seed can germinate in the time before frost hits.
And, when it does germinate, it has to grow a minimum height to stand up to what winter will throw at it.

With the clear understanding that any seed you throw on the ground now probably wont germinate at all, it will sit until optimum time in the spring when it will germinate.

The weather outside has only a bearing on what gardens, seeds, flowers, do....if the weather compliments the growth and since the tempteratures have been, and will continue to fall, the soil temperature is not conducive to seed germination.

SOIL TEMPERATURE is the deciding factor.....not air temperature.
In zones 5 & 6, it is generally up to September 15 when grass seed can almost guarantee decent results.
September 1st, is a better guideline date when considering planting grass seed. At that date, there is still ample time for the seed to germinate and live long enough to attain enough energy to withstand early frost.

LIME, being added in any amount, if put down without a soil test, is taking a chance. Lawns can be put at risk by over-liming just as much as overfertiizing.

The lawn company did a soil test? I doubt it. He can tell you the moon is blue and if you don't know the difference then the moon IS blue.
Soil tests are done by laboratories. Samples are taken from many parts of the property.....not just one place.

Ask for the test results of this test he did....on paper.
If he did a soil test that would have had to be done.

Liming a lawn is done at this time....and the results wont show up until well into next srpring; lime effects travels slowly through soil.

I've got alkaline soil, so I won't try to answer questions about liming.

When is your average first frost date? Generally, you want to seed 6 weeks or so before the average first frost if you want a successful fall seeding. Depending on conditions, you might be able to push that a bit, but not a lot. I'd expect frost to hit before early December in northern Ohio, so you're probably a bit late.

Your next option is to do a dormant seeding. Wait until it is too cold for the seed to have a chance at germinating (maybe before the first snowstorm) and put seed down. It will sit there until spring and work its way into contact with the soil when the snow melts and when the soil heaves with freeze/thaw cycles. In the spring, when the soil warms enough, the grass will germinate with normal spring moisture.

Thank you all for all the helpful information. If I am going to dormant seed, do I need to aerate or power rake first or will the weather over winter get the seed in enough contact with the soil to insure growth in the spring?
Jeanne, when I ask for the paperwork from the soil test what should I be expecting to see?
Thanks!

"Thank you all for all the helpful information. If I am going to dormant seed, do I need to aerate or power rake first "

No.