The question needs to be modified as “What always comes into a house through the keyhole ONLY?”

Answer: The KEY.

Clock / Watch / Time-piece…

That should be my NAME…!

No clause in the question confirms that the barber is a MAN, which implies that the barber can be a WOMAN as well. In that case,  she  can shave herself (if she wants to) because, as per the restrictive clause in the question, the ‘self-shaving ban’ is applicable to men only (“No MAN living in the town…). 

However, if the barber is a man, there is no possible answer because the question becomes a paradox in that case (due to the ‘all-inclusive’ first clause). 

That should be ‘B‘ – ‘b’cuase it lies within ‘AC‘ 🙂

COURTS

All other words are composed of same set of letters, viz., C, E, O, R, S and T.

You are Mr. ZERO….!

Two.

TWO

1. Outside and
2. Inside…!

For anyone who is chasing you, it’s YOU (who is harder to catch), and for YOU, it’s your breath...!