Can life exist on a asteroid? If the asteroid can sustain life, when the asteroid crashes on a planet can life survive going through a atmosphere and landing and then living on a earth-like planet it crashed on?
Asteroids/comets are typically thought to be the origin of prebiotic molecules (simple organic compounds) which aided in the construction of the first DNA/life on Earth. These molecules obviously were not destroyed during impact. If this is the sort of scenario you are looking for then yes, it is very possible.
However, it's unlikely that an asteroid could provide the heat required for life as we know it to form, so single-celled organisms would be unlikely, complex multicellular organisms out of the question.
See: https://en.wikipedia.org/wiki/Abiogenesis for more.
I would argue that Asteroids could in fact sustain life, at least in the bacterial form. In fact Ceres (OK technically a dwarf planet but piece would be asteriods) may have a liquid ocean beneath the surface . If there is liquid water then it has been stable for quite some time. At the very least Extremophiles would likely be able to live there.
Landing would be rough. Any asteroid that is large enough to sustain liquid water for long periods would deliver a fair bit of energy on impact. But if a smallish piece broke off the Asteroid while the big part misses the Earth and that small piece hits the ocean the bacteria might survive.
An asteroid could not hold active life (lack of atmosphere, low gravity, lack of stable heating mechanism etc.), but it could hold dead life even through atmosphere traversal and crash landing on a planet (if it is big enough to not get vaporized by the atmosphere) and could spread its inhabitants on the new planets. "Dead life" also includes eggs/seeds, so it could cause foreign organisms to live on a planet seeded by the asteroid.
In one theory, asteroids are though to have held basic organisms (we're talking single cell) that crashed and started life on this planet and possibly on others so yes, it definitely a plausible option. Nothing to advance though, don't expect any advanced alien races hiding out in one.