Yes. Film permits are required to film in public areas. For general state information visit http://film.ca.gov/. Permitting requirements vary depending on the locations and jurisdictions (i.e., federal, state, county and city), so be sure to also check with the local/regional film office. The California Film Commission has a directory of film offices found here: http://film.ca.gov/locations/regional-film-offices/
Contact the local tourism board and/or film commission for permitting information and applications using the directory linked above.
Contact the local film commission for rules and regulations, but plan to allow several weeks for processing to provide ample time.
This varies depending on the jurisdiction and location. Check with the regional film office.
For more Frequently Asked Questions (and answers!) visit the California Film Commission’s FAQ page: https://film.ca.gov/state-permits/permit-faq/