Animals sleeping is a pretty big science. Humans are the outliers on this, because we almost shut everything down while sleeping whereas most animals only shut down partially, and/or for much briefer periods. Our sleeping routine is very much based on being in a group, where someone would keep an eye out while others sleep.

Dolphins sleep with one part of the brain at a time. Some birds do the same, but if they're in a group, the innermost individuals can sleep fully, while the outermost keep one eye open away from the group.

As for sharks, yes, they do require a constant flow of water through their gills to breathe even when sleeping. Either they keep moving or they die, or as one shark was found to do: sleeping against a current.