Is there a special term for certain figures of speech that only make sense in a certain order? For instance, "sick and tired" is never said "tired and sick," at least not if the same meaning is to get across. Thanks!

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I'm not familiar with a term specific to this phenomenon. It seems to be simply a fixed expression that happens to have coordination (here, linking with and) in it. While you normally can switch the order of the coordinated items, since this is a fixed expression, doing so is impossible in this case.

Semantically, since the meaning is non-compositional (more than the sum of its parts), it can also be considered an idiom. Indeed, many idioms are also fixed expressions.

It is also of note that you could also think of it as a large adjective. (This is how it is treated in Wiktionary.) While historically it derives from three separate words, in the modern language it could just as well be treated as a single word and, e.g., written sickintired by an illiterate adult. Phonologically, you could think of this 'word' having a secondary stress on sick and a primary stress on tired.