Learn Dutch: Ik hoor niets met dit oor. - I hear nothing with this ear.

In Dutch, the word for "this" changes depending on whether the noun is a de-word or a het-word. Because oor (ear) is a het-word (het oor), we must use dit. If it were a de-word, we would use deze.

What is the difference between niets and niet? Could I use niet here?

Niets means "nothing" (a pronoun), while niet means "not" (an adverb). Because you are saying "I hear nothing", you need niets. If you wanted to say "I do not hear with this ear," you would use niet: Ik hoor niet met dit oor.

Is met always the right preposition to use when talking about doing something with a body part?

Yes! Just like in English, Dutch uses met (with) to show the instrument, tool, or body part used to perform an action. You hear met dit oor (with this ear), exactly like you would write met een pen (with a pen).