Question: Let say that the Earth's Moon was knocked out of orbit and landed on the Earth with some damage but not world ending. Would Earth's atmosphere envelop to include the Moon? Would the moon sink into the Earth? What would happen?

Answer: 
To this point, this is purely theoretical. We have no solid evidence of islands of stability. They may or may not exist to begin with. It's hard to speculate on the methods of creating something we don't even know exists.

That said, currently, the heaviest elements found in nature appear to be synthesized in the mergers of neutron stars. It might be possible that a similar process could create even heavier elements. It's possible that some of the discovered transuranic elements we've discovered might be formed in such mergers, however their half-lives are so short that they break down before they're detectable. If there are stable transuranic elements, then they might be formed in this manner, but we've never seen any evidence of them. Me might not know what to look for, but so far we've detected no suggestions of such elements.

Another option is the accretion disks of black holes. It is believed that nucleosynthesis occurs in accretion disks, and one could speculate that there may be conditions in accretion disks that could make it possible.

But without any evidence of the islands of instability, this is entirely conjecture.
The relatively well known process that is believed to produce most heavy atoms in the universe is the ["r-process"](https://en.wikipedia.org/wiki/R-process). The wikipedia page mentions s

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> . It has been suggested that multiple nuclear explosions would make it possible to reach the island of stability, as the affected nuclides (starting with uranium-238 as seed nuclei) would not have time to beta decay all the way to the quickly spontaneously fissioning nuclides at the line of beta stability before they absorbed more neutrons in the next explosion, thus providing a chance to reach neutron-rich superheavy nuclides like copernicium-291 and -293 which should have half-lives of centuries or millennia.
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It's not clear (at least from this source) whether this could also happen in a supernova or a neutron star collision, or whether other processes like photodissociation might destroy the large nuclei before they reach this putative island of stability. If they did reach it, it might be quite hard to detect:

1. Presumably only a small proportion of the nuclei would become that heavy
2. No one really expects the isotopes on the [island to actually](https://en.wikipedia.org/wiki/Island_of_stability) be stable, just to be more stable than "nearby" isotopes.
3. Since we don't have samples of these elements on Earth, we don't have accurate spectra that might enable us to detect them in space.