Quite all right! Many of these are indeed related, and they’re common questions one may have when they first get into raising meowth.
Regarding the first question, you’ll want to take your meowth to the pokémon center immediately. The coin is actually attached to its head, so if you notice that it’s missing, it’s either been torn off, or worse—it’s inside. Neither situation would be pleasant for the meowth (although for some reason, they’re less likely to notice the latter). Keep in mind that an embedded or ingrown coin isn’t deadly; it can just be painful, not to mention it opens a meowth up to infection.
With that in mind, unfortunately, you can’t really replace a meowth’s coin, and yes, the coin is actually created by the meowth itself, from birth. It’s really a false coin, if we’re being accurate—a little like a fingernail made of a metallic organic substance that just so happens to take the shape of a coin, but its bed is actually part of the skull. It’s only supposed to come off when a meowth is nearing evolution, as the coin bed is replaced by a jewel bed in preparation for growing out a persian jewel.
As for Pay Day, rest assured it doesn’t actually rely on the coin for use, although the coin is thought to be the means by which a meowth (or persian) channels this ability. Think of it as a lightning rod for the elemental energy needed to work. In actuality, though, meowth (or persian) simply channels their energies into normal elemental blasts that, in mid-air, convert into solid matter to form charms similar to the coins on their heads. These coins aren’t actually associated with any particular denomination (although they will always look strikingly similar to the money of a meowth’s home region, save for the lack of a date), but they can be collected and sold in the same way nuggets or star pieces can.
Or at least, that’s the legal way to do it. A lot of trainers either don’t realize it’s not actually money or do and attempt to use it as actual money anyway, to mixed results. As meowth coins are generally sold on a 1:1 basis (that is, they actually do have the same value as whatever amount of money they look like, even if they aren’t, themselves, actual money), eagle-eyed shopkeepers may offer to buy the coins or simply take the coins with a warning, depending on the age of the trainer. Stricter shopkeepers dealing with older trainers who should know better may attempt to accuse such a trainer of using counterfeit money. So, really, it’s important to know that Pay Day coins aren’t actually coins and must be traded for real money, not used immediately.