Are the words gamut and gauntlet interchangeable?

They are not. Gamut means the entire range or extent. Don't confuse this word with gauntlet.
Example: His reactions ran the gamut from rage to apathy.

Gauntlet has two meanings. First, it refers to a glove that in the days of chivalry was thrown to the ground to announce a challenge. Therefore, throwing down the gauntlet has become a figure of speech meaning to challenge:

Example: A fifty-year-old woman entered the marathon, thus throwing down the gauntlet before the younger runners.

Second, gauntlet (or gantlet) was also once a form of punishment in which a person ran between two rows of men who struck him. It has also become a figure of speech:

Example: In trying to promote her idea, she was forced to run the gauntlet (or gantlet) between those who thought it was too ambitious and those who thought it wasn't ambitious enough.