Sentence Construction – ‘You Idiot’ or ‘You’re an Idiot’: Which is Correct?

No. Remember the following:

concerned
adjective (WORRIED)
troubled with feelings of anxiety:
Sarah is very concerned about your safety.

concerned
adjective (INVOLVED)
​[not gradable] involved or involving:
They say that free trade will benefit all concerned.
Her job is only concerned with costs and fees.
(Cambridge)

The two examples given sound like you're worried about the car and the girl, whereas, without further context, my first impression is that interested in this car means you want to buy it and interested in this girl means you like this girl (or that you want to date her). Also, I think the preferable preposition in either case is about.

Concerned is listed as an adjective, so you cannot drop to be:

He was concerned with this car. [OK]
He concerned with this car. [Not OK]

You can drop to be when concern functions as a transitive verb.

This car concerns me.
This girl concerns me.

In either case, something about them worries you.

"Does", describing an action, does not need to precede "write", which, in itself, already describes an action.

An example of when does would be used is if the sentence is a confirmation/emphasis:

Wait, Mr. Johnson writes novels?

Yes, Mr. Johnson does write novels.

When expressing the past, write, the infinitive, can still be used, but does must be converted to the past tense:

Wait, Mr. Johnson wrote novels?

Yes, Mr. Johnson did write novels.