Rice – Korean rice cakes (tteok) sold unrefrigerated

Well, first off, mold grows from spores, and your Pyrenean cheese was likely already "contaminated" with Penicillium roqueforti, Penicillium glaucum, and Penicillium candidum spores at the cheese shop (it'd be surprising if the cheese shop isn't covered with them!). So, if you keep it in a environment habitable to them, they will grow. I suspect that you'd get the same results even if you put them in two different plastic containers.

As to its safety, if it was indeed one of the used-in-food Penicillium molds, it should be safe to eat—you can make sheep's milk blue cheeses (e.g., roquefort) after all. But the problem is that the same humid environment that encouraged the Penicillium colonization also encourage other molds—many of which you don't want to eat. Also, I'm pretty sure there are blue-green molds you don't want to eat.

Based on your edit, my best guess is that this particular rice has a really high amount of surface starch, which gelatinized when you first tried to cook it (liquid from the mushrooms might have been enough to get this result). That would have created a moisture-resistant layer around the grains that inhibited moisture from soaking in further. Your soak would have washed that away and allowed the rice to properly absorb the broth.

That's definitely weird - probably a byproduct of how this particular brand gets processed. A further experiment worth trying would be to briefly rinse off the grains in a colander for your next batch. That would also get rid of surface starch without quite the same effect as a soak. If they work after even a brief rinse, I'd consider that pretty good evidence for this theory.

Another thing you could try might be sauteeing the rice in a completely separate pan from the mushrooms, so that they're exposed only to oil and the surface starch gets toasted instead of gelatinized. Maybe not quite as desirable from a flavor perspective, and of course you'd have to clean another pan, but it's worth testing if you really want to figure out the root of the problem.