You still draft the card that you remove. This may matter if an ability refers to "the next card you draft" or similar. If that card instructs you to draft it face up, do so, then turn it face down as you remove it. If that card instructs you to reveal it as you draft it and then perform additional actions, perform those actions before turning the card face down and removing it from the draft.

A card that is removed from the draft isn't in any player's card pool. That card can't be played in any games associated with that draft.

You can't remove cards from the draft that you've already drafted. Only cards drafted after you draft Cogwork Grinder may be removed this way.

If you draft multiple Cogwork Grinders, the value for X will be the same for each of them. It doesn't matter that some of the removed cards may have been removed before you drafted the second Cogwork Grinder, for example.

No other player may look at the cards you remove from the draft (unless Cogwork Spy allows a player to).

If a Cogwork Grinder owned by another player enters the battlefield under your control, the number of counters it enters with will be based on the number of cards you've removed from the draft with Cogwork Grinders, not the number its owner removed. Unless you also drafted Cogwork Grinder, this will be 0.