I pretty much have the same questions as OP, but with an addition:

2b)Should I rake after core-aeration so that more sunlight is available for seeds to germinate?There seems to be conflicting opinions on this....

If you want to keep the kinds of grass you now have, just overseed with the same grass (Zoysia on one side and KBG on the other). If you want to replace one or the other, you'll want to kill the existing grass on that side before seeding.

Do you seed zoysia? It seems that any time I've seen ads for it, they're advertising plugs.

I don't know the answer on fertilizer timing.

Syman--There's no need to rake the cores unless you have low spots and you're trying to use the cores to raise them (slightly).

quote--->"Do you seed zoysia? It seems that any time I've seen ads for it, they're advertising plugs."

That's what I thought too. Anyone know if there is Zoysia seed?? I may just aerate it and only overseed the KBG side...but I'm still up in the air about the fertilization question.