is there a fee to get in the park?

Yes. And one to visit lighthouse

Yes, $10 at the gate however it's free for Volusia county residents December and January

No

There is a fee for entry, I don't recall the price. Details should be on their website.

$ for cars, can walk in from free parking outside park or beach if you've paid to get on beach

Yes, 10.00 but park outside the park and walk in for free