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It certainly sounds like you take top-notch care of him!
How old is your cat?
What color is he?
Are the dots raised at all, or just darkened areas of skin?
I admire your concern for your cat! This time you can breathe a sigh of relief. I almost asked you in my first info request if he was orange, because the cats I see this on are mostly orange cats (it can also occur in calicos and tortoiseshell calicos).
This condition is officially called 'lentigo simplex' and is very common and benign (ie, doesn't do any harm). The common term for them is just freckles. They don't do any harm, but usually don't go away either, and just as you've discovered, won't wash off.
They can also pop up on the insides or outsides of the ears, so just be aware of that, and it's still nothing to worry about in that location either. There's also no increased chance that your cat will develop anything bad because he has these freckles.
If you notice more of them, as long as they all are flat and appear like the others, are located on the lips/gums, eye lids, nose or ears, I would not worry. They seem to pop up quickly just as you've described.
If he ever develops a spot that looks different -- raised or pendulous instead of flat, pink instead of black, etc -- have that one checked out. But again, he's no more likely to develop anything bad than any other cat.
Please let me know if you want to discuss this further. I"m happy to answer any more questions you may have or clarify something I've said that doesn't make sense. Just reply to me to ask more questions. Otherwise, best wishes for a wonderful Thanksgiving! I've included a picture below...let me know if you can't see it.
Here's a picture courtesy of Veterinary Information Network. It's of a person holding a cat's head and ears back so you can see the pigmented lips and nose -- the cat looks a little creepy but I think the picture demonstrates lentigo well on the lips, nose and eye lids. Does this look familiar?