That depends on a lot of factors. One of the advantages of life being on a planet or a reasonably large moon is the presence of an atmosphere. Atmospheres allow gases that may be important for the survival of life to stay close to the surface of the planet or moon. Additionally, atmospheres block some of the harmful cosmic radiation. It’s also important that the asteroid would have the organic molecules that are important to life as we know it.

Water in its liquid form is also thought to be critical and scientists don’t think liquid water would be on an asteroid. Life may be able to survive harsh conditions, and indeed we’ve found life on the outside of objects sent into space. However, there is a big difference between surviving in harsh conditions and growing. Some organisms can go into a sort of suspended animation state where they barely need anything and can thus survive the cold vacuum of space.

However, they can’t eat or reproduce in that environment so in that sense they aren’t carrying out the basic functions of life. So it’s possible that if a particularly hardy organism was put on an asteroid that it could survive the harsh conditions, but it is extremely unlikely that the same organism would grow, reproduce, and evolve on an asteroid.

Good question. Life in or on an asteroid would be more difficult than in or on any planet, because any water would have to be trapped inside of crystal inclusions and unable to escape. There would also need to be a source of energy for life that didn't get exhausted. This latter one is particularly unlikely in all but the very largest of asteroids like Vesta and Ceres, which are arguably small planets (Ceres is even classified as a 'dwarf planet', like Pluto). For these reasons I suspect that the answer to your question is "no", at least, not without help from intelligent life from elsewhere. However, one of the things I've learned as a biologist is "never say never", so I won't say that life can't survive on an asteroid.