What's the difference between
"I wish you a long happy life."
"I wish you a long merry life."
I am especially interested about the cases when these adjectives are applied to a long period of time.
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We would use "merry" to describe someone's character or behavior (merry laughter/merry nature/merry eyes). It's also possible to use "merry" to emphasize something that you are saying, often when you want to express disapproval or humor (It hasn't stopped the British Navy proceeding on its merry way). You might be thinking of the greeting Merry Christmas. I'd say that in this case "merry" is used in a kind of old-fashioned way. According to the Collins Dictionary, a merry sound or sight makes you feel cheerful.
But we don't usually use "merry" to describe someone's life. Mind that in British English "merry" can also mean slightly drunk. So, in some contexts it may be OK to say "long merry life", but it wouldn't mean the same as "long happy life". Rather, you'd wish someone a long life of a person who likes booze. This may be not the only interpretation. Remember context is everything.
According to the Online Oxford Collocation Dictionary, "happy life" is a natural collocation. But the dictionary doesn't say anything about "merry life". So, after all, it's not about a short or long period of time. It's about what the adjectives in question usually describe.