comets - How did water get on Earth

The key finding why we think Earth's water came from Asteroids (big rocks) and not Comets (small rocks) is the Deuterium/Hydrogen ratio that we can measure in several sources.
When a star forms, it has an initial value of D/H that came from the nucleosynthesis in its progenitor nebular / star.

In a protoplanetary disc, as dust grows to rocks grow to planets your nebular gas will be trapped with initial D/H ratios in gas giant atmospheres. But the way to get Water onto Asteroids and Comets (those don't have substantial mass to retain atmospheres themselves!) is sublimation and maybe adsorption.
The latter two processes are strongly sensitive to the gas-mass and therefore different D/H ratios from the protosolar ones are expected.
And in fact we find them to be different:

This was in the news lately, as ESA managed to touch down and measure 67P's D/H ratio which gave another hint on the asteroidic origin of water on earth.

This finding, however does not resolve the question:

• There maybe other isotopical tracers like D/H that give a hint on the history of water.


• There could have been a decrease in D/H and subsequential increase again, or the other way round.


• We know that the protosolar nebula must have had massive amounts of water (Oxygen is the third-most abundand element in normal stellar fusion!), so again the question is, why did Earth retain so much water, while the lighter planets Venus and Mars retained the heavier element $CO_2$
  ...

I could continue this for quite some time, but the bottom line is: We only have hints, not definitive answers.

To refer to the rest of your question: A hot plume in a colder atmosphere will not necesarily leave the whole gravitational potential well.