Do stars actually generate muons directly? From what I understand the muons on Earth are a result of cosmic rays colliding wtih particles in the atmosphere.

Muons are naturally generated by cosmic ray protons colliding with atmospheric molecules and creating pions, which then rapidly decay to muons and muon neutrinos. These themselves then decay into a bunch of other things.

If they do, how far do they travel before decaying? Even if they travel at relativistic speeds, they have a mean lifetime of 2.2 ns, so the math seems to say they don’t travel very far at all on average.

That muons can hit the Earth is one of the key pieces of evidence in favor of relativity, in fact. As you say, with a mean lifetime of 2.2 nanoseconds, they shouldn’t be able to hit the surface of the Earth, but because at relativistic speeds time dilation occurs from our frame of reference (or, equivalently, in the muon’s inertial frame, it sees the distance it has to travel be radically shortened via length contraction), they do end up hitting the earth.

Either way, are there any other sources of muons in the universe? I’m curious what the muon density distribution in the universe would look like.

I doubt it, because they decay so quickly. AFAIK you have to do it via the pion decay route, and all the muons we create are in particle accelerators. I guess it would be like how we create radioactive isotopes in hospitals on-demand for medical purposes that wouldn’t survive transportation to the hospital before decay, and couldn’t be stored long-term because, well, they would decay.

as an aside, Nature is rather more pessimistic about the discovery, which I think is reasonable.