Yes. Grasshoppers are a member of the Order Orthoptera, within the Class Insecta, Phylum Arthropoda, Kingdom Animalia. Remember the generic description of insects, that they have six legs.

Yes, grasshoppers are their own type of insect; the locust is a type of grasshopper. Grasshoppers are related to crickets.

As a supplement to ariel's handy taxonomic answer: the order Orthoptera also includes crickets, katydids, and locusts. Many of them produce sound by rubbing their wings against each other or their legs, a process called "stridulation". Most of them have powerful back legs designed for jumping.