I'm going to give you my opinion. This is based on years of farming, and also from my experience seeding grass. But, I'm afraid I'm just going to be just another opinion.
1.) NEVER fertilize and seed at the same time. The fertilizer can kill the germination of the seed. Always wait till grass is growing to fertilize. (I suggest in the spring.)
2.) I don't de-thatch.....ever. I think it's a waste of time if you are going to core it. Or if you aerate.
I'd mow it short and sweep the lawn. (No clippings to inhibit the core aerator.) Then core it. Then seed it. (Because for whatever reason, seed seems to grow better in the ground.) The next step is choice, I think any of the following is ok: Either use a dethatcher to "scuff" the dirt/cores to cover the seed. Or you could use a spike aerator. Or, use a lawn roller. No matter what, end with rolling it with a lawn roller. Doesn't have to be heavy, just something to firm the seed bed. Plus, it will help break up the left over cores. Then water it everyday. I don't know what kind of soil you have, but I use a grass seed blend from my local (farm) seed store. It's a blend of fescue grasses, rye grass and Kentucky Bluegrass. It does very well because the seed that grows best in the different soil/sunlight/shade/dry areas thrives, and the other grasses die off. That way, you get a uniform looking coverage, and don't have to seed differently. I pay about $150 a bag for the seed.
That's my take on it!
