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To get 20 BTU/hr per square foot the floor would need to be 10F above room temperature. Let's say 1.5 inches of wood is R1.5, so you'd need the plate to be 30F above floor temperature to get that 20 BTU/hr. So if the room is 72F the plate needs to be at 112F. The water needs to be slightly warmer, not a lot, so 120F sounds about right. Maybe a couple of degrees more because you need the average to be around 120.

There are plenty of hardwood floors heated by radiant. you just have to watch your p's and q's.

Everyone will or should tell you that you need a proper design for your system. Depending on the system you and your contractor decide on, the manufacturer will have engineering data that will tell you if their system will work given your design outdoor temperature and the amount of insulation in your house. The data will also tell you how to program your boiler to deliver the proper temperature water to your floor.

Most staple-up systems should not exceed 120F water temperature as the flooring can react at higher temps. Look also at the Ultra-Fin product as you can exceed 120F since this product does not heat by conduction, rather convection. I like it because you only need one run per bay instead of two.

https://www.ultra-fin.com/

Yes, floor temps should not exceed 80-85F as it gets uncomfortable for humans. You start sweating at those temps.

I have aluminum plates on the underside of the underlayment under both hardwood and even carpet/pad. Not a problem. On a reset control with 120 degree water on design day. Runs most of winter in the 90-110 range, p/s, long cycles.

A thermal imaging camera makes balancing much easier.

You need to start with knowing what the insulation will be and doing a heat loss calculation, then you or someone else can use some radiant design software to design tubing to cover that loss.

You will have to insulate under the tubing to push the heat up through the floor.

It is done all the time with hardwood. I don't know what the limits on the surface or the tubing are. Not only will designing for too high a surface temp dry out the wood, it will also be uncomfortable to walk on.

Underfloor heat needs to be designed, there are no useful rules of thumb. It is possible for it to be the sole source of heat, it's also possible for it not to be, it depends upon your heating load and the layout of the house.

I would look at a system like Warmboard (not an endorsement, that's just the brand name of the most popular system).

Will you have air conditioning? It often works much better to deploy the heated floors strategically and supplement with forced air.

Floor thickness won't be an issue with doors, you can raise them. Where it may really be a problem is with stairs. Although if you do both floors you might be able to raise the entire flight by the thickness added.

Are there any more controls like this one? this is possibly the High Limit control but the setting is at 120° which in my mind is too low to be the HI limit

This photo indicates that your heating boiler is located near your clothes dryer, and that clothes dryer vent is allowing lots of lint into the boiler room area.  All the telltale dust on the gas valve tells me that there is most likely dust on the actual burners inside the fire compartment, and the combustion air openings of the burners.  That dust can lead to insufficient combustion air and incomplete combustion.  That can lead to the boiler heat exchanger getting a build up of soot and carbon that will cause more problems that you do not want to deal with, some day this winter.

This shows that you have three 24 volt transformers.  You only need one to operate that heating boiler.  Without seeing the bigger picture I have to ask how many thermostats you have?   That may explain the reason for more than one transformer.  

But getting back to your wiring problem, If you have only one thermostat for the entire home, then at least one of those transformers is not for the heating boiler.  So I am guessing that the door bell is connected to the transformer on the left   That leaves the upper transformer with a relay, to operate the circulator pump, and the other transformer on the lower right looks like it has the same 3 wire that is near the high limit control in the top picture. 

Now let's just say that you get a professional to come to make sure the 1949 boiler is operating safely and properly and that professional is not experienced on older model equipment. They might tell you that you need new equipment (and they would probably be right when it comes to operating cost) by telling you that this old stuff is unsafe. (that would be wrong information) and you didn't mind the "operating cost" gas bill which you feel is very affordable, and therefore no amount of fuel savings will justify the cost of a replacement boiler.  You will need to know how to put the wires back…

I am only guessing at this point as to what you might have… but this is a safe wiring diagram that will shut off the gas burner in the event the water in the boiler gets hotter that the High Limit Setting. I would like to tell you the put the red wire here, and the green wire there but you only have gray wires, so I have placed colors on the diagram for clarity, but electric is color blind and will travel thru any color wire. I have made the assumption that the top transformer/relay is connected to the thermostat. When the thermostat calls for heat, the relay in the silver box will connect 120 VAC to operate the circulator pump. There will also be a connection in that same relay using low voltage (24 volts) to make the gas valve operate the burners.

This is most likely where you have the three disconnected wires.  I have indicated that one of the three wires is Red and connects from the transformer to the R terminal on the aquastat.  The W on the Aquastat will get connected to the TH on the gas valve indicated by the white wire in the drawing.  Then the TR on the gas valve will get connected to the transformer  indicated by the blue wire on the drawing. 

I hope this helps.  And I really recommend that you call a professional to check your boiler.  It is probably in need of some professional maintenance. Hopefully you will get someone experienced that will actually do a good job for you.

Mr. Ed

By the way,  when I did gas boiler maintenance I never found it necessary to remove the wires you removed.  This might tell you something…

Take a photo of the burner control to which the wires were attached… and post that. And then figure out where the other end of each wire is and what it attaches to and post that. Someone will figure it out ( @EdTheHeaterMan is a genius at that!)