Not my area of expertise so someone may correct me later on, but the prevailing theory is that all life is descended from a single abiogenesis event, mainly because any life we have found evidence of has functioned exactly the same (well, basically) as things alive today. No one sticks out enough to say "yeah, that's funky, that doesn't work the same as everyone else"

I highly recommend Life's Edge by Carl Zimmer for an interesting read on this topic, because a lot of the questions you're asking depend on the very-much-disagreed-upon definition of "life." There are still scientists looking for signs of abiogenesis, in labs and in the field (particularly around volcanos, hot springs, thermal vents, etc, because that's what Earth used to be like).

>Let's assume that the nanites are made of carbon

>the mass of the Moon is 10^22 kg. So one Moon is more than enough to build such a ring around the Galaxy...

Carbon is only available in trace amounts on the moon. Carbon makes up about 0.46% of the universe, but it's not like it's conveniently all in one place for mining.

I think you're confusing multiple different concepts here. I'll let someone more versed in information theory weigh in on the black hole discussion, but ashes themselves are just the non-burnable parts of whatever you set on fire.

In a log, for example, the vast majority of the mass is water (which can evaporate into the air) and organic compounds that will be turned into carbon dioxide and carbon monoxide (this is the chemical reaction you are doing when you see fire). After you completely, 100% turn the organic compounds into gasses, you are left with ashes, which are largely the micronutrients that were stored in the plant and used to keep it healthy. Calcium, potassium, magnesium, iron.