In the situation you are describing, the only thing one can take away is that the room you are in is brighter than it would be if the bright room were not there. It does not mean that your room has to be as bright as the bright room, just that your room is brighter than it would be otherwise.
For your new situation with the street lamp:
If you move very far away, but still within sight of the lamp, the light from the lamp or nearby bright objects will still reach your dark room. Your dark room will not be as dark as it was before, even if you judge it to be just as dark using only your eye. Since light is reaching the dark room, that light will cause the room to be a bit brighter.
Using the idea of intensity of light:
An intensity of zero at some point in space means there is no electromagnetic radiation; there is no light. If you stand in a place where the intensity of visible light is actually equal to zero, there is no light there. You won't be able to see anything with your eyes or any other visible light detector.
Make sure you're not confusing zero intensity with a very small intensity. These are very different ideas.
It might help if you take a particle-view of light. Think about the photons leaving the lamp and entering your eye. If they enter your eye, and you move away, they're going to hit the walls of the room instead.
Incandescent bulb are black body emitters. Basically, something is heated enough so that it radiates most of the power put into it. The black body radiation spectrum is well known, with it being a function of temperature. The lower the temperature, the more the bulk of the radiation shifts to lower wavelengths.
Normal incadescent bulbs used for lighting are actually horribly inefficient. We don't have a material that we can heat up so hot to maximize the amount of visible light in its black body radiation spectrum. Actually, we can heat material that high, but it won't last long. We don't have a material that won't evaporate or sublime slowly enough to make a useable lightbulb with its black body radiation optimized for visible light.
Real LEBs (Light Emitting Bulbs) are a tradeoff between efficiency and lifetime. The hotter you run the fillament, the more efficient but also the shorter it lasts. This effect is quite non-linear. A little hotter makes a big difference.
There is therefore another reason to have separate heat lamps from ordinary LEBs, which is to make a more economical IR emitting bulb. The fillament operating temperature is adjusted for good IR production and less light. This does increase efficiency a little, but a ordinary LEB still emits most of its radiation as IR already. However, running the fillament cooler greatly increases bulb life and forces less tradeoffs in the design otherwise. Not only does the fillament evaporate slower, but the wire can be thicker, for example.
So dedicated heat lamps are a little more efficient at their intended purpose, but also last a lot longer and are more rugged and durable than incandescent bulbs shifted more towards producing visible light.
Best Answer
There are two aspects to be considered here:
For both of these sub-questions, the answer depends on the material that the wall is made of and on the wavelength of the infrared radiation. At wavelengths close to the visible (for example, at 0.8 µm), the behaviour is likely similar to the visible — so similar, in fact, that there are meteorologists that refer to any radiation up to 1 µm as "visible", even though we clearly can't see such radiation. At 20 µm, however, behaviour may be vastly different than at 0.5 µm.
The exact answer depends on too many factors, so the answer is: it depends. If you can specify exactly what material the wall is made of, and in what wavelength you are interested, a more precise answer can be given.
For example, for the walls of the Ice Hotel, mid-infrared radiation (10–20 µm or so) is certainly absorbed a lot more than visible light!