Regarding hot vs cold sake, I don't think it's expensive vs cheap. Cold sake is a pretty modern development. Traditionally it was served warm. In fact, in the Japanese historical dramas they show it being served from teapot-thingies. Most of my middle aged Japanese-immigrant friends drink theirs warm, but that might be generational.
Light-bodied, dry sakes do best chilled, while heavier bodied, sweeter ones are the ones to have heated. The big thing to remember is not to overheat, as that drives off some of the flavor.
I'm not convinced that omakase is a great idea unless you and the chef have a good rapport. Too much risk of getting things you don't enjoy.