Can I use my birth certificate (united states, broward county) to travel on the ferry and return back to Fort Lauderdale without any issues?

Yep. I have done it twice. Have a state issued photo ID with you as well. make sure it is the state certified birth certificate and not one from a hospital or other non-certified origins

No one must have a passport

You need a passport to reenter the us.

Its best to call, but i think state id s are no longer accepted and you will need a passport to travel international

Yes, you should be able to.