Legal Question in Business Law in California

No. The original of the demand is kept by the propounding party. A copy of the demand gets served on the party to whom it is directed, and a copy is served on all other parties who have appeared in the action. (Code Civ. Proc., sect. 2031.040.)

The party responding keeps a copy, and serves the original on the propounding party. The party demanding the inspection retains the original and the original of the response until six months after the action has been disposed. (Code Civ. Proc., sect. 2031.390, subd. (b).)

The inspection demand and the response are not filed with the court. (Code Civ. Proc., sect. 2031.290, subd. (a).)

No. Not only don't you "need to" file it with the court, the rules say not to file it.

If you and your opponent get into a dispute about the discovery you may need to submit a copy as an exhibit to your motion or opposition. You may need to use these papers as exhibits in other contexts, too. But you should not file them on their own.

No, discovery in general is "between the parties" and doesn't get put on the court's file unless and until there's a dispute about compliance with the discovery rules which requires the court to see what was asked for and what was (or was not) furnished in response.