After Scarifying Questions

Hi all! Hoping for a bit of advice.

Lawn had very very little maintenance over the past 5 years or so and was very mossy and generally in poor condition.

I bit the bullet earlier this week and did a pretty deep scarify. It’s obviously in really poor condition now which I was expecting, but just hoping for some advice on what to do next:

1. How long should I wait now before overseeding?

2. Some of the grass is very wet and often in shade. Should I add any sand (or anything else) to try and alter the very clay- like consistency of the soil in these areas?

3. How much grass it put down? I presume I’m pretty much looking at laying a new lawn on some areas here 😂

Thanks for the help!