My house in Minnesota had a similar set-up.
The duct to the mechanical room is code required for make-up air for your (gas) furnace and water heater. If you do not have gas or oil (i.e., no combustion) it is probably not required. My utility room was reasonably well sealed and it was not a space conditioning problem for me.
I added an outside air connection to the return air side of furnace to pressurize the house to mitigate smokey air infiltration through the damper on my exterior wall fireplace. Pressurizing the house helped a bit. It did not have a noticable effect on utility bills. A building with positive inside pressure will have less air infiltration, so while you are heating cold air, it may help in other ways.
A HRV is what you need. You may also be able to fit gravity back draft dampers in the ducts, although I do not know if code permits dampers on for the make up air.
Bruce