What's the difference between худой and плохой? Are these words interchangeable? When should each be used?
One of the obsolete meanings of "худой" is bad, but nowadays this meaning is almost non-existent as an adjective. The only form that preserved that meaning is "худо" which is indeed similar to "плохо".
You can say "мне худо" instead of "мне плохо" - though it will sound slightly bookish.
You can not say "худой" instead of "плохой" in modern Russian apart from some idioms like and sayings like "худой мир лучше доброй ссоры".
"Худой" stands for thin, skinny. Actually it always was the main meaning.
Плохой is neutral word for 'bad'. Худой is neutral word for 'skinny, thin' (of a person).
But худой can be used in the sense 'bad', according to the dictionary (where it's marked as традиционно-народный - folk style, traditional, used for proverbs and folklore) - I wouldn't say it's obsolete though, but characteristic of that particular functional STYLE (folklore connotation).
So these two words are really not interchangable.