There were still bombardments by large asteroids occurring, and some years ago I recall reading how one paleobiologist speculated that life actually tried to start perhaps 3 to 6 times before it got a firm toehold. Each time, it was wiped out by a devastating impact far larger than the one which may have occurred 65 million years ago, wiping out the large dinosaurs. I personally believe that the synthesis of life on the Earth is an indigenous and inevitable process given the chemical soup that no doubt existed on the Earth then. Chemists have a ridiculously easy time of artificially creating pre-biotic organic molecules from the primordial atmosphere and water of the ancient Earth. There have been speculations that most of the organo-synthesis occurred inside comets that also bombarded the Earth 4 billion years ago, but I have a major quibble with this scenario.
Chemical reactions occur at a rate that depends on temperature.
Any cometary reactions at temperatures near 20 - 100 K, would take MUCH MUCH longer than identical reactions on the surface of a nice warm planet that just formed. Although astronomers can look into distant interstellar clouds and see astonishingly large molecules, these are still quite small compared to the amino acids needed for pre-biotic RNA and protein synthesis. The comets may have transported a rich molecular soup frozen onto dust grain surfaces, but such a soup could easily have been synthesized on the Earth in a lot less than the millions of years taken by the chemistry of interstellar clouds and cometary bodies. Scenarios that attempt to synthesize whole DNA molecules inside comets have an even tougher time doing so because of the low temperatures involved on the surfaces of cometary dust grains.