My assumption is, that you refer to a rendering based radiosity technique.

1 - As it is right now, any light that is visible, it will contribute to triangle color, even if it is kilometers away.

One idea I had was to use black fog, distance based, so distant lights/objects would gently fade to black, contributing little or nothing to the color.

Is this the best way or are there alternatives?

There are at least two ways. If a light emitting surface is far away, it will automatically occupy only a few pixels, therefor its contribution will be quite low. But a fog algorithm could help too to limit the rendering distance (performance reason).

2 - What if I want a spot light, how to make only objects in the cone of the light get affected by it?

Or a point light, only affecting a certain radius?

Rendering the light as 3d cone/sphere would not work, unless I am missing something?

There are several ways. I would implement a first, light emitting pass, where you render your scene from the light perspective and distribute the energy on the visible surfaces. This way you have control about the light distribution method (ie spotlight, sun etc). The next pass will then switch to normal energy gathering (rendered from the perspective of a lit surface).

True, distant lights would appear small, just a few pixels, I did not take that into account.

Can you explain better what you mean by rendering from the light perspective?

I dont understand how to apply that to render a spotlight for example

One way to do radiosity is to render a half-cube (render the scene 5 times, approximation of half sphere) from the perspective of a single polygon/surface and sums up all the energy of the visible surfaces (in form of single pixels).

The other way, quite useful as starting point, is to render the scene from a light perspective (6 camera , full cube) , but instead of rendering any surface energy values (color) you render an ID of the surface (ie. encoded as unique RGBA value or directly as 32bit integer). Then you sum up all visible pixels for each surface (refered by ID) and propagate the energy of the light source depending on some function (ie. angle, light cone etc. to simulate a spot-light) to the according surface.