Basement – the best flooring for the basement

From a comfort and water protection perspective, I think you'll find it beneficial to install a sub floor. I'd recommend one which allows for ventilation beneath it such as these 2x2 panels you can buy at Home Depot/Lowes

Dricore

This will give you a nice, warm "softer" surface to put your carpet pad on, and help protect from any condensation issues with the cold floor beneath by allowing it a place to form, and then with the proper ventilation openings cut into the subfloor (standard floor vent in the carpet to allow airflow) it'll dry out on its own.

My bottom line would be that you want to do anything you have to in order to get the standing water out of there. That may very well require some work to redesign the outside drainage lines. I'd look into an outside drainage system that fed into a nice deep rock filled pit that allowed excess water to disperse into the ground away from your house foundation. You may also want to equip that outside pit (also known as deep french drain) with a sump pump to take care of extreme heavy rain or high water table incidents. After the drain situation is corrected I would look strongly at correcting the landscaping and roof runoff gutters to you minimize the amount of water that the drain pit is asked to handle.

All that work can be done outside without tearing up your floor in the den. Hopefully the area under the floor would then stay dry like it should.

As far as heating....trying to use a soil and concrete "tunnel to bring heat into a room it terribly inefficient. I would find out how you can disable that part of your existing heating system. Then have a suitable sized direct vent heater installed in the room for in room heating. A direct vent heater sits on the floor by the wall and has a concentric pipe system goes out through the wall. The heater draws combustion air in through the outer cool pipe and exhausts through the inner hot pipe. I've installed one of these in a basement room that was more than half under ground and had good luck with it. Obviously you have to route electricity and a gas line over to the heater if you have gas type heating fuel (either natural gas or propane).

An advantage of the separate heating zone created like this is that you can easily regulate the temperature to suit that specific room or just shut it off it the room is not in use.