Is Toad's head a hat or part of the body?

Solution 1:

According to this video, it is a hat. They show a scene from the Super Mario Bros. Super Show where Toad is being flown away by some sort of dinosaur-like creature (forgive me if that's some sort of Mario character). Toad manages to escape, and used his head piece as a parachute, thus revealing that it's really a hat, and not his head.

The Mario Wiki also references that this is a hat, under the Super Mario RPG: Legend of the Seven Stars section:

Many Toads in this game have different patterns on their hats than normally found on Toads.

Even the Wikipedia page for Toad has a picture of him with his Mushroom Cap/Hat removed, however it does mention that none of the games seem to confirm or deny this:

Toad has on occasion removed his mushroom cap, revealing three strands of hair; however, the games have never confirmed or refuted this.