He would have joined because the Empire was a dictatorship, as well as trying to rid the galaxy of Jedi. Dooku believed that there was corruption in both the Galactic Senate and the Jedi Order.

Dooku would have been the next General Grievous.

Grievous served under Dooku. Dooku would have served under Vader.

Maybe, maybe not. There are differences in situations, of course. Even Sidious had decided that Ventress, who was more of a Sith than Grievous, needed to be eliminated.

Yes, because he was a puppet.

I say most likely since he was the most loyal ally to Palpatine and when he would have learned The Empire being formed he would have sided with his master