If i am understanding the layout correctly the physical installation would be very simple. My only concern would be how big the space is, and whether this additional load would upset the internal pressure balance of the ductwork.
If the point of discharge is only 8' from the place you intend to tap you should be able to use insulated flex duct successfully.
You could put the register in the wall directly below the door in the manner you describe.
The next consideration is return air. You cant push air into a room unless there is a return path for the same volume of air. Once you have heat into the room you should be able to leave the door open as a return path, otherwise you would have to install some type of return duct or register, which could go into the adjacent room.
You stated that there is a 1' step down, that is really tooo much and you should consider adding a step, especially coming off the laundry where peoples vision is often blocked by carrying laundry baskets.
