Determining which good sentiment to wish at each holiday

Happy Christmas is fine in British English, but it does sound odd in American English. As Peter Shor says, for most other holidays, "Happy ---" is fine.

There's no rule; they're all idiomatic. Even though 'merry', 'happy', 'jolly', and synonyms are, well, synonymous, it doesn't make them interchangeable. People just use patterns over and over again to the point where the minutest change sounds strange (this should be part of the definition of 'idiom'). Sometimes such patterns arise 'organically' (Out of the common ether), and other times one person says something clever, in person or in media, and people start copying it.

As to alternatives for Christmas and other festive holidays, one could use good, great, or wonderful. They are both innocuous and non-idiomatic enough to not sound too strange:

have a good Christmas

have a great Thanksgiving

have a wonderful Easter holiday

As to holidays in general, correct, not every holiday warrants either a 'happy' greeting or even a special greeting at all, and only culture will tell you what is appropriate. Many religious and political holidays (Good Friday/Tisha b'Av/Ashura or Memorial Day) you just show up and commiserate.

Of course, by cultural analogy with the juggernaut of Christmas, many 'special' days seem to warrant a special similar greeting (like Happy Halloween where historically there never was one). There's no necessity here at all, it's just what people like to do and like to copy in others.

I used to think that "Happy Christmas" was merely the result of people saying the politically correct "Happy Holidays" and then switching back to "Christmas" but not replacing the "Happy" part with "Merry".

However, there's a John Lennon song with the title Happy Xmas (War is Over), which was released well before the PC trend.

Incidentally, the Japanese have imported "Merry Christmas" into their language, but don't use the word "Merry" in any other context.