Teardown labor is free if you DIY. There remains the cost of the disposal though.

What do you have for flooring?

It might affect resale if you teardown the room.

Luxury vinyl

Not enough info, but this sounds like a capture of an old concrete patio - a very common DIY occurrence - and comfort is only 1 problem compared to a host of others. You would have to give info on your existing heating system, how this is built, etc to understand if you can, or even should, make it seamless to the rest of your house. Unfortunately, when you talked to each contractor they are only concerned or experienced in their trade, not really looking at the big picture.

If it seems to be in reasonable condition, I would add a few more windows, making sure all had screens, that can be opened. I would also add a door into the house, and then use it as a nice screened in room, with real windows, just to sit and read, and enjoy, when weather permits.

Situations like this are invariably, insulation. Floor, walls and ceiling/roof. Don't do anything with heating until you've taken care of this.

I had a similar situation in my house...extension added on concrete slab with no underfloor insulation, and walls poorly insulated. Insulating the walls properly and replacing windows, as part of a larger project in our case, made a big difference. The room was still colder than the rest of the house, due to the cold floor and probably inadequate HVAC registers. But it was much more tolerable.

The flooring material has little effect on making a room cold or not.

Personally, if I didn’t really need the space, I would have it all ripped off. You say the room is too cold to use and it’s probably increasing your utility bills for no benefit.

Unless, of course, I had a bunch of spare $$ to dump there.

" Even in summer it's cold! "

I'd love to have 3 or 4 rooms that are cold in Summer. But then you might be stretching it right?

"I've talked to HVAC guys, flooring guys, insulation guys, and electricians about baseboard heaters. "

I am curious about what your bids were for improving the insulation? I don't see what the flooring people can do. There's not much point in adding heaters if the room isn't properly insulated, so you are left with insulation. Since you are already paying (apparently) to heat the room and it's not retaining the heat, then it's the heat loss from the walls/windows/ceiling.

Maybe your local electricty/energy company can send someone out for a consultation. Have you called to ask?

Otherwise, I think you are looking at insulation to solve a lot of the problem, and perhaps improving the windows.

I think I figured out the issue.  The sliding door.  It was covered in frost from this cold we had.  BTW, I did add thermal curtains as suggestions from some people I knew.  I finally got around to closing them and that is where the frost came.  It was everywhere, and along the frame.  I did replace one window in this room recently, so I am going to have to replace this sliding door.  I really hope it solves part of this never ending chill.  Didnt know how bad a window or sliding door can be.  Maybe I've been spoiled by all the places I lived in.

Purchase a thermal camera and scan the room for heat loss. Make repairs as needed.

"BTW, I did add thermal curtains as suggestions from some people I knew. I finally got around to closing them and that is where the frost came."

While air leakage at your sliding door may be contributing to the heat loss, the accumulation of frost after closing the insulating curtains is likely due to the effectiveness of the curtains. By limiting the transfer of heat from the room to the glass of the sliding door, they caused the interior temperature of the glass to drop below the adjacent air's saturation temperature.

I agree with G&S that the proper approach is to get your hands on an infra-red camera and use it to identify problem areas. You can get IR accessories which work with your smart phone for under $200. Once the problem areas are addressed, you'll need to evaluate whether your existing HVAC system is capable of conditioning the space.

Some Home Depot stores have infrared cameras to rent. Before you replace the door you should really ID the air leakage in the entire room. If the rest is really bad, spending thousands on a door might be reconsidered.

Thank you everyone!  I will look into the camera!  (I do have to replace this sliding door one day.  The tracks are pretty bad from neglect.  It doesn't glide well and sticks.  I tried sprays and it only helps so much)

I actually just took a thermal thermometer around the room, it has a house setting.  The sliding door is about 40 degrees around.  The walls about 59, and the further away I go from the sliding door it gets warmer.  I also checked around the other door and window and I can remove the trim and add loctatlite.  My front door was low 50's, but I am getting that replaced.

You might also find that carpeting would be better on that floor than the vinyl you have now. Put in carpeting with a very good pad under it. Sure will make your feet a lot warmer.

Yes, wall to wall carpet is an insulator. It makes a big difference. We have w/w in the basement gameroom and livingroom room at camp. Both have concrete underneath. Concrete floors are a cold that keeps on giving.

Carpet prevents bare feet from being in direct contact with a cold surface but the R1-R2 insulating ability of carpeting will have little or no effect on the temperature in the room if the area below is cold.

Do the obvious things:

Check the outlets & switches that are on outside walls. Take the covers off and see if foam gaskets have been installed. Are there gaps between the boxes and the drywall. If so, caulk or use gap & crack foam to fill. (Wear gloves and use masking tape & paper to avoid getting the foam on the finished wall outside of where you want to fill. Some cardboard on the floor to protect it.

Check for drafts coming from under the baseboard. Either use caulk backing rod shoved under the baseboard or remove the base and caulk where the wall plate meets the floor.

Pull a piece of trim from a window or the door. Apply gap & crack foam to seal between the window/ door jambs and the framing if that area is not insulated.

Check any penetrations in the wall & add foam to any gaps there.

Do the same for any penetrations in the floor. Plumbing, HVAC vents. Seal as needed.

Drop any ceiling light fixtures and seal around the electrical boxes.

Post a picture of the outside of the extension where the slab meets the ground. It is sometimes possible to insulate the outer perimeter of the slab and the footings.

ETA: Have a look see if the attic insulation is adequate, too.

Thank you!  I will have to do the work around the trim, and use foam.  All the outlets have that insulation.  I tested all this today with that thermometer and was finding the areas with cold spots.  Biggest was the sliding door.  I covered it and it has really helped!

I cant use carpet in this room because of the entry points in this room.  It would get dirty and trashed.  However, what really keeps the feet warm in that room is the foam gym flooring.  I bought some at Costco last year, around new years.  I'll be picking up more when they are back in the stores.  I had carpet padding under the area rugs and this foam gym floor is much better than carpet padding.  I can walk on it bare feet and not get cold.  

Thank you everyone for bringing my sanity back down!