What fraction of starlight, seen from Earth, is actually reflected light? Thanks to reflected starlight, many planets and comets in the Solar System have been visible to humans since long before the development of modern astronomy.

Some of the starlight from outside the Solar System should be reflected light as well. That is, light emitted by one star in the Milky Way may encounter another star, and be reflected toward Earth. Normally, stars are assumed to be perfect blackbodies, but in reality they must bounce off some radiation.

Also, starlight should be reflected off exoplanets, moons, asteroids, meteors, comets, brown dwarfs, etc. What fraction of starlight, seen by the naked eye from Earth, is actually reflected light?
A truly negligible amount. You only need compare the brightness of the planets as viewed from the Earth with the brightness of the sun. A very rough calculation (considering the relative magnitudes of the planets and the Sun) suggests that light emitted directly by the sun is 100 million times brighter than light reflected off planets.

You note that the sun is not a perfect black body, and so does reflect some light. However it only reflects light reflected on to it from the stars, which are a very weak light source. And most starlight that does fall on the sun is absorbed. I haven't tried to estimate how much of the sun's light is reflections of starlight, but it is a very very small proportion. I'd guess much less than a billionth.

There is one situation where reflected light is significant, and that is when very bright stars illuminate dust and gas, to form a reflection nebula. These are common and at least part of the orion nebula is reflected starlight. Other reflection nebulae are too dim to see.

While a I don't have an exact figure, it is clear that nearly all starlight is directly emitted from the star, and it takes special equipment to see any reflected light at all.