Their advice is right, birth certs are fine. Just make sure they are govt-issued, not hospital ones, and not xerox copies of the originals. And the teens will need a photo ID of some sort, preferably govt issue (like a driver's license.) If they are not old enough for a license, you can use a school ID or something similar, in case you are asked to provide one for them. Hassles at the port occur because 1) people have the wrong form of ID, or 2) they get an overzealous and/or inexperienced agent to check them in, who has been told what ID is needed, and won't budge an inch from the exact spec's they were told.
As far as excursions - best to check out the port reviews on this site, and also past posts on cruise-chat. And don't be afraid to get off the ship in Roatan, assuming you are going there. It's a beautiful island, English-speaking, and has lovely beaches. I'd recommend excursions through the ship in most places, though, just to make sure no one gets lost and left behind! Have a great time!