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.

Citations: http://en.wikipedia.org/wiki/Orthoptera

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

Citations: http://www.uky.edu/Ag/CritterFiles/casefile/insects/grasshoppers/grasshoppers.ht...
http://www.absoluteastronomy.com/topics/Grasshopper

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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.

Citations: http://en.wikipedia.org/wiki/Orthoptera

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