No and No. I lived in CB, just north of the pier for several years. I surfed just about everywhere there was to surf and also did a fair bit of fishing off of Canaveral. The island and surf zone are off limits and you really wouldn't want to surf them when compared to other/better spots. They shoal pretty badly, resulting in small, often confused breakers (much like the point at Hatteras, only way smaller). In both cases, what looks like a point on the map is really just a wide-open sandbar. I'm sure that the "never say never" principle could apply. I've seen Cape at Hatteras crazy good, big, almost scary... but never saw Canaveral resemble anything close.
No, you cannot. You can only surf there if you have access.
It's not worth it 99% of the time, anyway. Like Philos guessed, it's shoaly and shallow. Crummy surf on the beach and crummy surf on the shoals. If the cape is good, it's probably much better elsewhere.
Dont let them fool you. Old Skip and Gumbo used to surf there in `99 till the government cut them out (disability checks). Nowadays only the few in FL with boats can access the spot, pretty much solid breakers 89% of the year.