I'd make a $5 bet that if they find and sequence life on Mars it'll be not just speciously like archaea but actually archaea.

> I wonder if some extreme events could have kicked up enough microorganisms that some even went to the moon!

We could fit 56 planet mars in row in the distance between the moon and the earth, so... not. Really unlikely

Fun little fact: you can fit pretty much exactly all the planets in the Solar system between the Earth and the Moon. More precisely, the eccentricity of the Moon’s orbit is small, about 5.5%. The sum of the diameters of all the planets is a number in the interval between the lowest and highest distance between Moon and Earth, i.e. between 363k km and 405k km.

We find rocks from the Moon and from Mars[0] on earth, and have reason to believe that small pieces of rock travel from most inner solar system bodies to most others.

I find it totally believable that some micro organisms could survive the ride on such rocks and possibly thrive on other bodies, if the environment is suitable.

[0]: https://en.wikipedia.org/wiki/Martian_meteorite

Rocks expelled and melted after a collision. If i'm not wrong, none alive or organic would survive on magma. Not even prions.

Depends on how fast the magma cooled. There are life forms which can handle up to 122°C and still actively reproduce. Look up hyperthermophiles for more info.

When the magma has cooled enough the rock would be in space or in the upper mars atmosphere at least, not touching anything.

Anything growing on mars surface would burn, only an hypothetical floating microbe could land in this rock after cooling, but would be like expecting a bacteria landing in a bullet, surviving the impact and then sticking to the surface whereas still alive (and much later surviving a second impact against earth after years in outer space). Seems unprobable.