It depends on where you want to use the phrase. If it is a subject, you should say they and I. If it is an object, you should say them and me. The example you gave should be the only difference between them and me is this, because them and me is the object of the preposition between.
word choice - Is it: 'They and I?' Or: 'Them and I?' Or: 'Them and me ...
"Are they?" is the standard question form. In a question, we normally begin the sentence with an interrogative word like "where" or "how", or "is" or "are" or "do" or "does". "They are?" is an example of a declarative statement turned into a question by simply putting a question mark at the end. In speech, it is intoned as a question, i.e. you raise the pitch of your voice at the end of the ...
Could someone please explain to me the correct order of using he, she, they, we, I, and you as subject in a sentence. Can they all be used in one sentence like example 3?
grammar - Order of he, she, they, we, I, and you - English Language ...
They are really cute simply asserts that your particular dog and cat are really cute, and doesn't require any more context than the two sentences you've provided. Those are really cute would only make sense if the preceding conversation had already introduced the possibility of having different types of animals as pets.
I suspect when people use they for a country it means they are thinking of the people rather than the country. You could argue whether it is the country or its inhabitants which experiences a recession.