How to get rid of gnarly roots in an outdoor French drain without hiring/renting a roto rooter

You can find the end, or at least the first leakage point in the line, with one of these:

It's intended for cleaning drain/sewer pipe. How it works is the small bladder/balloon will expand under the water pressure of the hose (garden hose water source) and has a small exit point on the other side of the balloon. So it will expand to the pipe size then fill it up. This way will use a decent amount of water but on a dry yard day should work fairly well.

Item/Picture source but they come is all different sizes and should be available at your local hardware store. I would measure the pipe first and buy the largest/closest size one you can, this is a 3" to 4" pictured but it may not seal a 4" pipe well until it has a certain amount of back pressure.

Same concept different approach, you can get 4" (or 3" depending on what you have) pipe fittings and step it down to a garden hose fitting.

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.