I can give you a logical "proof" for both laws: Current law: if the sum of the currents into a node were NOT zero, then there would be current either coming in from, or going to into, nothingness. That would not be science, it would be magic. Likewise for the sum of voltages around a loop.
Depending on where your physics is at, you may know that voltage is really potential difference. There is an electric potential field with a scalar value defined for all space (including within your circuit wires) related to the work done to push a unit charge to that point from a pre-agreed point: usually infinity. The voltage between two points is just the difference between the potential at each point. If the potential difference between point a and b is v1 and the potential difference between b and c is v2, the potential difference between a and c is v1 + v2. As an alternative approach, consider v1 to be the work required to push a unit charge from a to b and v2 the work from b to c. Clearly, the work to go from a to c is the sum of the two numbers. Hope this helps
Afaik, his law of radiation is another conservation law. It says that the emissivity of a surface is the same as its absorptivity. A consequence of this is that two surfaces at the same temperature, facing each other will each absorb the same radiated power at any wavelength as they will emit. So neither surface can heat up at the expense of the other. That would generate free Energy - verboten!
Is there, precisely, a ? I have read around what I can find in a couple of text books and on the www. It seems as much an article of faith as anything, based on the second law of thermodynamics. You may have to dig further than just what Google has to offer at the first level. And it's not just a matter of a mathematical derivation. This link refers to the original paper and this link has a lot about the studies at the time. It points out the big error in Kirchoff's approach. Hope it's of some use.
The Wikipedia article discusses Kirchoff's Law of Radiation in some detail: http://en.wikipedia.org/wiki/Kirchhoff's_law_of_thermal_radiation