DW and I are pretty new divers, so we don't have BCs, regs, or wet suits yet. We did bring fins, boots, masks, snorkels, and our computers along. Personally, I like having my own kit...if we had the rest of our gear, we'd have brought it with. Most of the people in our 2 dive excursions (Cozumel and Grand Cayman) did bring full sets of gear. They seemed to be primarily either adult couples or with one small kid, so space wasn't as big of a concern.
Now...4 full sets of dive gear, that could get a bit cramped. Might be difficult to fit all that in the room and still have space for people.

We kept our stuff on the veranda most of the time, both to let it air out and to not take up room in the stateroom.
Not sure when/where you're looking to go, but we didn't need wetsuits for either dive in SEP. I'd check out the local dive reports for the month you'll be there to see if you'll want exposure protection. Boardies and a long sleeve rashguard were plenty for me.
That's quite incorrect. Aside from tanks and dive knifes, scuba gear is most certainly allowed on-board.