Live rock pretty much provides the most surface area for nitrifying bacteria to grow and assimilate on. That nitrifying bacteria is what will convert the toxic waste your inhabitants produce into less harmful nitrates.
So yeah, live rock is pretty essential to the saltwater aquarium. Even fish only tanks that don't have live rock in the display tank have some sort of device that provides a large amount of surface area for bacteria to live on, such as bio-balls, filter pads, and even live rock in the sump.
And yes, you can cycle your tank with a piece of shrimp. That's how I have always cycled my many tanks. If you get uncured live rock though, you probably won't have to add the shrimp since there will be plenty of die-off from the live rock to produce ammonia in order to get the cycle going.