Intrinsic value (or fair value) is a meaningless concept when applied to currencies. The US dollar has value because the US government maintains a fairly effective monopoly on violence over a large chunk of real estate. They require residents to pay taxes in that currency, and if you don't pay then the government will seize your assets by force. Overall this system works pretty well.

You're close. Fiat money has value when it's backed by something. The US$ is backed by the US government. It's still fiat money.

It is "backed" as in endorsed, or supported, by the US government. In the same sense that Bitcoin is "backed" by its many worldwide supporters and miners. It is not "backed" in the traditional meaning of being redeemable for something else at a fixed rate ("backed by gold", "backed by silver", etc.).

The intrinsic value of fiat money is that I can pay my taxes with it and avoid persecution by the state.