How much starlight do black holes absorb? Can it be calculated how much light from other stars a black hole of certain sizes would absorb? And how long will it take for the number of active stars to decrease to the point where Hawking radiation would surpass it?
tldr; extremely little. Space is big, even biggest black holes are ridiculously tiny in physical size

The amount of starlight absorbed by black holes is really minuscule. A black hole is not going to be sucking light in: it can be fairly well approximated to be an absolutely black object around the size of its event horizon (or photon sphere, does not really matter for the purpose for this thought experiment, since photon sphere is 1.5x EH). 

Now, let's think of occlusions by more familiar objects: the Moon and the Sun. Now let's think our Moon would be a black hole. Retaining its current size, it would need to have mass in range of about 500 solar masses - pretty big for a BH. It would be also orbiting really close to our Sun; so close, it would probably get eaten by the Moon-BH relatively soon. Also, we would be observing this system really close.

But even in this case, we would still see the Sun most of the time unobstructed. Obstruction means absorption of light, thus there's really not much of this happening. 

The amount of absorption of light from a star can be calculated by figuring out the ratio of black hole surface area to the surface area of a sphere with the radius equal to distance between the star and the black hole. In most cases you will be using square kilometers for the BH area, and square light years for the sphere. Which is almost exactly the same as comparing single atom mass to grams. 

For the second question, star shine does not really contribute much to the temperature of space, so CMB will dominate this until very very far in the future, when all the stars are already gone