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Not talking about the defective, naughty "Pre-Gremlins" that you get when water hits them,... but actual natural reproduction.

Does it occur only, like, every 100 years?

Are there any female Mogwais in existence? (Keep in mind, Gimzo is explicitly referred to as a "he", and we saw a distinctly female Gremlin in the second film.)

Or, much like the rules governing their general care, are there some specific circumstances that have to allow for a Mogwai to reproduce naturally?

I mean, they couldn't just pop into existence, could they?

I think that all indications are that they reproduce parthenogenetically, precisely by getting wet. You are being thrown off by the fact that the other mogwai produced in Gremlins when water got spattered on Gizmo were much more mischievous and dangerous than their relatively mild-mannered progenitor. However, the in-universe reason for the difference in temperament seems to be that Gizmo is just a really, really friendly mogwai; more typically, the mogwai are not willing to get along with humans or serve as pets.

The out-of-universe reason that the later mogwai, created from getting Gizmo wet, were distinctly nastier, even before they turned into gremlins, is that the movie was originally going to be much darker. There would have been a lot more blood and violence, including Mrs. Peltzer being decapitated by the mogwai. There was no "good" mogwai (the name means "demon" in Cantonese) in the original treatment, and the original creature would have become the gremlins' ringleader, Spike. When the plot was changed to give the film a somewhat lighter tone, the personality of the progenitor mogwai was toned down, and the creature became the friendly, goofy Gizmo.

The second film features all sorts of new, modified gremlins (The New Batch was its subtitle), produced by the creatures consuming genetic modification serums in a laboratory. So the form taken by any creatures in that movie is not standard. Out-of-universe, the Gremlins 2 was only got produced because the makers were allowed almost total freedom to make the second film a spoof of the first.

Mogwai can't reproduce. They're genetically engineered creatures, at least according to the film's official novelisation.

Mogturmen, the inventor of the Mogwai species, had seen to that. Centuries ago on another planet, Mogturmen had set out to produce a creature that was adaptable to any climate and condition, one that could easily reproduce itself, was gentle and highly intelligent. Exactly why Mogturmen embarked on this venture is not known, except that such inventors flourished during an era of widespread experimentation in the field of species creation—an era, it should be added, that passed into disrepute following later, unsuccessful attempts to introduce cross-pollination among certain species of crawling carnivores.

At first, Mogturmen’s experiment had been looked upon as a great success and he was hailed as the genetic hero of three galaxies. The first sets of Mogwai turned out as planned, although the gentle little beasts had a few drawbacks not foreseen by their creator. Their vast intelligence seemed to interfere with their ability to communicate (Mogturmen said it was because they thought so much faster than they could verbalize), and for some unaccountable reason they were repelled by light. Discounting these deficiencies, the galactic powers ordered the Mogwai sent to every inhabitable planet in the universe, their purpose being to inspire alien beings with their peaceful spirit and intelligence and to instruct them in the ways of living without violence and possible extinction. Among the planets selected for early Mogwai population were Kelm-6 in the Poraisti Range, Clinpf-A of the Beehive Pollux, and the third satellite of MinorSun#67672, a small but fertile body called Earth by its inhabitants.