(1) The rocket fires under water: that works because it's a true rocket, carrying both fuel and oxidizer on board and not using air at all. The shape of the nozzle is also important, since water has an amazing ability to absorb heat and thus quench combustion; a caul around the nozzle keeps the water from getting at the ignition point from the sides.
(2) The primary explosion creates a roughly circular set of cracks. Those lead to "big" cracks opening up in the ice, as existing seams and weaknesses get expanded.
(3) Note that the circular cracks become hexagonal (matching the big cracks) as you move outwards. What's happening is that the inner cracks happen first, and then those start the big cracks, which actually spread faster than the round ones, basically moving at the speed of sound in ice. By the time the last round cracks are forming, they're forming in ice that already has hexagonal cracks in it.
(4) Water is an incompressible fluid, which means that the shock waves are going to hit things under water very, very hard. This is likely to be lethal to anything underwater and nearby, and is how dynamite fishing works. The reasons why that's a terrible idea (and is therefore illegal nearly everywhere) are hopefully obvious.
(5) That said, watching things explode underwater is pretty cool.