No, I think he's talking about the initial, chemical explosion that sets off the fission + fusion part, at least in the old bombs.

That's right. Note that there are other Z-pinch fusion designs which use the Z-pinch to directly trigger fusion reaction without going through a fission intermediary.

Really? Water is just about the best radiation shield in existence. It would only affect the ~10 feet of ocean immediately around it.

It's not about the actual radiation, that's easy to deal with. It's about the containers cracking or leaking and having that stuff floating around in the ocean, contaminating our food and the environment. Even the strongest container we can make is still susceptible to the corrosive effects of the ocean.

If you're going to bury it in the ocean, your best bet would be to bury it near a subduction zone. But then you have the earth itself possibly being the cause of the container cracks and leaks.