Tried this last night. Findings! (this presumably applies to all large fish)
1. Any cage will work fine for keeping the carp from drowning. I caught mine by digging under a pond full of carp, littering the floor with cages, then digging up into it. All but one fell through and into the cages, though I don't think there's a way to make this process work in a non-luck-based way. Anyhow once the water levels died down to 2/7 and 3/7 my dwarves ran and stored them outside.
2. Dwarves can swim/walk freely in 3/7 and below water (the 4/7 is a myth, they won't do it), but refuse to go into 4/7 water. Carp drown in anything less than 4/7 water. Therefore it is impossible to have 'meeting' points between the dwarf half and the carp half. This makes having a 4/7 room a waste, and it a better tactic to just put the things into 7/7 water.
3. It is absolutely possible to have an accessible 4/7 (or any water level really) room above a 7/7 tunnel with other open spots - just use a baffle (diagonal tile) to keep the taller room from depressurizing. The tunnels will still be perfectly accessible. However due to point 2 making a room like this is purposeless.
4. Carp cannot be war/hunting-trained, but can be tamed easily. You can tame a creature fine while it's caged, but for advanced training you need to drag the pet to the kennels, which is impossible with the water differences.
5. Carp cannot be adopted. Even a dwarf with swimming raw'd in forced to stand beside the carp will forego regular activity because the water is there. And if the dwarf is not in the water with the carp they cannot see/path to the carp and thus will not adopt them.
6. Carp are boring as shit. Because underwater meeting zones don't allow meeting zone room, carp will not path to hang out in the water where you want them to. They will in effect sit exactly where they were released FOREVER.
Ultimately all these findings mean while you can absolutely FARM carp, you can't do anything remotely useful with them aside from taking advantage of their petvalue of 50 (which I don't think is all that high). HOWEVER! I think there's still a way to take advantage of them for defensive purposes. As carp are known to fight back fiercely when they cannot run (ie in a pond), you can pit carp into 1x1 pits of water all along your entrance to serve as 3x3 trap zones - the carp has nowhere to run and thus will attack anything in the 8 land tiles surrounding it - and the enemies it attacks won't be pathing to it save maybe a lucky angle from a marksgoblin, so it'll have the surprise advantage. A goblin will be walking by and then suddenly dragged into the water, where it will be ripped apart and unable to defend itself because it will be drowning and focused on getting out of the water. I haven't tried this but I believe this would effectively make your carp murderous weapons of mass death and actually serve as an incredibly effective defense, and easy to set up as well. The next step is to set this up and see how it works in practice.
My friends, I believe the carp has successfully been weaponized for dwarven defense.