Can you live in Caesar Head park?

There's no campground or hotel next to Ceasar's Head. There's an option to stay at Flat Rock State Park campground and drive from there (about 30 minutes)

Nope! There's No Camping allowed at Cesar's Head State Park

No.

There are a lot of private homes that were on the mountain before the park was acquired by the state. Some of the houses are 100 years old. Those home owners technically live "in" the park but on their own land.