If you're in the front of the power curve, your drag will also
increase with the higher airspeed.

Yes.

Unless your nose-up trim change upped the angle of attack...

Yes.

Do you use the door trick to induce more drag and increase your
descent path as if you're you're adding more flaps?

No, the door thing is to bank the aircraft. I presume it deflect that
air upwards toward the underside of the wing, increasing its AOA.