Are any of the phrases in the title incorrect in any way? Do the meanings differ in any way? Is one preferred over the other and if so then why? I find this particularly interesting for many reasons including because a listener can not distinguish whether a speaker said "There they're." or "They're there." Also, are there other more complex and/or more interesting examples of this kind?
“There they're” is nonstandard English. Contractions of be, had, and would cannot appear at the end of a clause. They can only appear as unstressed sentence elements, and inverting the sentence order in this way (called stranding) puts stress on the verb. This avoids any ambiguity with “They're there,” for this particular case.
Just as it changes the grammatical stress, inverting the word order also changes the sentence's semantic stress, emphasizing there instead of they, answering slightly different questions: