Set Drawing Mode to Combo Alias. Click Tools > Doors (or press F6). Click a door direction and decide what door type. When you return to the Door main dialog box, the mouse cursor disappears!
Also while you are placing warp return points, item points, Link spawn point, stairs, or flags while you are still in Combo Alias mode, the cursor disappears when it reaches the screen and cannot reappear in the lower dialog box, the combos on the right and the Menu bar on top. (You will still see an X as the cursor with the Alias on the screen and when you click, the alias will NOT be placed. You are placing a point or a flag instead)