You, yourself, the real body will still activate dr.
Only the attack used by the mirror image and the tree will not activate dr. (to ensure you don't kill yourself by the mirror image attack when they attack about 0.2 seconds after you attack.
Speed doesn't work in jump quest such as mechanical heart.
But snow shoes that you can buy from El Nath armor shop will increase your acceleration/deceleration to the max even on non-icy surface (doesn't improve max speed).