According to Wiktionary there is a noun nuisance tax. Does this suggest nuisance can be an adjective? Is it?
What you have there is a noun adjunct.
In grammar, a noun adjunct, or attributive noun, or noun premodifier is a noun that modifies another noun, and is optional—meaning that it can be removed without changing the grammar of the sentence; it is a noun functioning as an adjective.
Nuisance is not an adjective in the strictest sense. Nuisance tax is what is known as a noun compound, and the first word in that phrase is known by a variety of names. See this question that talks about the variety of names (one is noun adjective, interestingly).
Other common noun compound examples are space shuttle and computer programming.
No it doesn't. Unfortunately, though this is a slightly unusual usage of "tax".
Normally, an X tax is a tax on X; income tax is a tax on your income, sales tax is a tax on the sale being made, window tax was a tax on the number of windows your house had, and so on. In all of these cases, X is of necessity a noun.
I hadn't come across the term nuisance tax before and Etymonline doesn't list it, but the definition you link to suggests that it derives from the tax being a nuisance itself. Despite the change in meaning, this is basically the same grammatical form as all the other taxes.