In the sentence, You sat down:
You is the pronoun subject
sat is the verb
down is the adverb.
There is only one noun in the sentence: student
I sat down.
I sat down on the divan before laying down on it.
I sat in a chair while watching television. Bob slowly sat down, but was still curious!
Me and Mandy were cold, so we sat down by the fireplace.
I personally find "the idea of" superfluous.
The adjective in "She sat down hurriedly at the spare desk", is the word "spare", describing the noun desk.
He was Walking down the street and He sat on the curb.
The violist sat poised and ready for her part to begin.
No, sat is a verb - the past tense of sit. You can begin a sentence with a verb if it is an imperative sentence but then the verb is in the base form not past. eg Sit down! Look out. Clean up your room.
Sit is already a verb because it is an action.Other verbs are sits down, sitting down and sat down.Some example sentences are:"I will sit down here"."He sits down over there"."I like sitting down"."He sat down next to the window".
The cat placidly sat down on the chair and slept.