Are game shows legal by nature or is there a catch that allows them to be legal? Are they considered a sweepstakes? How is one able to give away prizes without it being "gambling?" Do the game show contestants have to agree to some sort of contract to be involved? Is it a state by state issue?
Game shows are not gambling because the participants do not wager their own money. They risk the show's money only. (Also as @aroth points out in the comments, the vast majority of game shows can also use the "game of skill" defense.)
Yes. Game shows use contracts for contestants. See the following examples.