I'm not sure about whether “What's past is past” is well-structured or not. Please help me to find out the structure of it. (I wonder if it has double passive clauses; in other words, is past here a verb or an adjective?)
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It is fine, as is the extended "What is past is past."
There are no passives. The main phrase is "Something is past" with the verb is followed by an adjective. That something is the clause what is past, again with the verb is followed by an adjective.
If you wanted two passives then you could write "What is passed is passed" or "What's passed is passed." Spoken, these sound the same as what you wrote.