As one of our former members used to say, this isn't "internet-able." I don't see any way that anybody here can determine how the place was built, or how to go about possibly improving its comfort and utility, without looking at it in person and probably opening things up to have a look inside.

Did you have a pre-purchase home inspection? Did the inspector note anything about this room?

If it's unpermitted, that can give you headaches when you go to sell the house. At that point, you may indeed be looking at demolition. Whether it's safe or practical to use as a home office until then, I can't say, and I can't even suggest who locally might be able to advise you. Sorry. I hope someone else here has ideas.

An infrared camera can detect missing insulation in the walls and ceiling. If it is uninsulated, blown in or foam insulation can be added without completely starting over.

Since you bought the house last year, can we assume your local municipality did not object to the addition to your house? If so then improving it should not be an issue. You may still have to apply for permits depending on what you do.

It is hard to believe someone would build an addition with a door, windows and electric service outlets and install no insulation within the wall cavity It possible, but highly unlikely. If this was built with a permit application, then ask you code official if he has any drawings showing how it was built. It may have information about insulation.

Trying to bring in the duct work from the main house through exterior brick is not an easy chore. I think this would be a good application of a mini split system. A single head could service the whole room. You would have an independent thermostat to set the heating and cooling. You can set back the thermostat and save money when you are not using the room.

It’s not permitted as a habitable space, because that required heat. Its going to have elevation issues too if you I want to try to get an insulated floor. My garage has drywall and windows and no insulation. As did the 3 season room. That was pretty much the as built norm for those types of spaces. Removing drywall to fix the sins beneath is is actually the easiest and cheapest route to go to start down the road to this being safe, comfortable, and legal to occupy.

Check the walls with an IR thermometer as suggested above. If there's no insulation in the walls, get some blown in. Waterproof and insulate the slab (thick foam and carpet are probably best for this) and then see if you can get an HVAC duct and return put in. It won't be a "legal" bedroom but you can certainly use it as one.