BTW many flooring materials can be heated, including engineered hardwood (according to some of the manufacturers). Decide what material you want on the floor and do some research - chances are whatever you choose can be had in a variety that is capable of being heated. The Nuheat thermostats I have used in my home have a switch on them to limit the output for wood / other flooring. With stone/ceramics etc. you can heat them up quite high - 90+ degrees - but you don't want to go that high with wood etc. In terms of how long it takes to feel the heat, what you want is a programmable thermostat. You tell it what temp you want the floor at what time and it will get it there for you. Don't turn it on and off manually - you'll always forget and it will become a nuisance rather than the luxury it is. I use my heated floors in the two bathrooms as the primary heat source and it works great in Toronto, with a 2nd floor bathroom with 2 exterior walls in a drafty 90 year old home.

Yes you can put radiant under hardwood (or carpet or tile or...). We did using Zmesh from Heatizon. I think the bigger problem is the slab. Engineered HW would be better for over concrete. You don't say how high your ceilings are or whether you already have a heat source in that room. We chose to install the electric radiant system because we have a very open plan with 26' ceilings. Forced air w/vents at the top of the wall was completely impractical. We installed it as part of a huge remodel and we specifically designed the system so the transformers would be hidden in a mechanical room. You might want to do some research on electric radiant systems and the feasibility of installing it in that room considering what the electrical and mechanical needs are. Wanting to install it under hardwood shouldn't be your deterrant.

I would probably do something to trip a CO detector and then complain to your landlord. I am a bit surprised that any of this would be legal in NY which has some very tenant favored laws. 400 watts isn't going to do anything. Your best bet is an electric blanket and a down coat. It sounds like you have no option to have safe heat.

We have wood look tile on most of the main floor. We put it in 18 years ago and I still love the look. The larger stone look white tiles with light grout, I don't like as much because the grout has been a pain to keep clean. Slate look tile also ages well due to dark grout. Pros of tile are that it is hardy, easy to deal with dogs and clean ups. Hard to damage. Cons are hard to replace if damaged, grout is annoying and cracks at the expansion points between room pours of the gypcrete. And it is hard on your feet. If you get tile make sure they have a expansion line at the doorways that has flexible caulk and not grout. For grout get the premixed kind that doesn't require sealing. Better yet get epoxy. We found the premixed kind cracks less. The problem with tile that spans across pour areas or from one room to the next that is on a different heating loop is that it expands and contracts at different rates, thus cracking the tile and grout at doorways. We have had two rounds of carpet. The first included radiant floor rated rubber pad and low pile berber carpet. This was perfect and transmitted the heat well. When we replaced it we got the current version of the radiant floor rated rubber pad and a thicker plush carpet. The plusher carpet does not transmit the heat as well. The same manufacturer rubber pad (just their current model) stunk for 5 years of rubber. It was really bad at first and got better with each year but was just noxious. We had the manufacturer rep out and of course he couldn't smell it. For years we aired out the house at every opportunity and kept those bedroom doors closed 100% of the time unless we had all the windows open. It was strange because the first version of this same pad had no odor at all. We also have tarket fiberfloor in a bathroom. It is soft and easy to lay but it is easily damaged so good for low use areas. If I could go back in time I would have used a sheet linoleum like I had in my old house. It was a solid surface (color all the way through) sheet flooring and not the printed picture under a piece of thin plastic type. It may have been commercial grade but I'm not sure. I got it as a remenant and the installer loved it so much he stole the left overs to use in his own house. We didn't put it in because we didn't think it would be high end enough looking. But after dealing with cracked tiles, grout cracks, grout coming up and stinky carpet pad for years I would have loved to have gotten what was practical and let the next owners worry about looking posh.