The triggered ability is put onto the stack after you have already drawn your card for the turn.

If Howling Mine leaves the battlefield before it resolves, then the last known tap or untap state of the card is used for resolution.

It does not trigger at all if this is tapped at the start of the draw step, and it checks this again on resolution.

The additional draw is separate from any other draw during your draw step. It happens when the triggered ability resolves.