Don't aerate after the seed is down. Aerating first will expose more soil and give the seeds more area to germinate. Aerating after seeding will pull some seed up and crush other seed.

Thanks. When is the best time to fertilize the seed/new grass and is top dressing really helpful?

You could, however, do a final pass with the tines up to improve seed/soil contact.

You have a big project, and it would have been better if you started a few weeks ago.
Don't know what your entire lawn looks like so cannot advise whether to do just the brown spots or overseed the whole lawn.
I would aerate, seed and fertilize the same day; as soon as possible.
The most important step is to keep the seed moist for a few weeks.
You may want to do the worse half this fall and the other half next spring or fall.
It depends how much time you have to devote to your lawn.
I never top dress but I am retired and can keep the soil moist all the time.
I am in the second week of renovating about 12,000 sq ft.
It is very time consuming to keep every thing moist if it is sunny and breezy every day.