English doesn't even go up at the end of sentences for all questions, just yes or no ones.

Vietnamese doesn't. The rising tone that you hear at the end of an English language question can change the literal meaning of a word in Vietnamese.

Um Chinese may want to have a word with you

So the pets recognize it and respond.

They don't.

I was expecting Mandarin to be an exception, since the language uses pitch to encode different words; apparently it isn't, the speakers simply "abstract" the phonemic vs. phrasal pitch variations as two different things, when interpreting the sentence. Check figure 6.

And while there is a particle overtly conveying "this is a question", ⟨吗⟩ /ma⁰/ (the "0" indicates neutral tone), it seems that you can couple it with an assertive phrasal pitch to convey rhetorical questions. And other languages (like e.g. German and English, that overtly mark questions with verb fronting) show a similar pattern.

I also found some literature claiming that it might be cross-linguistically consistent

The most important observations are the following:

1. pitch tends to decline from the beginning of an IP [intonational phrase] to the end, a tendency known as declination;
2. the beginning of an IP may be marked by a local sharp rise in pitch or “reset”;
3. in IPs that are utterance-final and/or in statements, there may be a local drop in pitch at the end of the IP in addition to any overall declination spanning the IP as a whole;
4. in IPs that are in questions and/or are not utterance-final, declination may be moderated, suspended or even reversed, i.e. the overall trend may be less steeply declining, level, or even slightly rising;
5. in addition to exhibiting reduced declination, non-final and interrogative IPs may also have a local rise in pitch at the end, or at least have no local drop.

The validity of these observations, as general tendencies, is not in doubt.

The article also lays out some potential explanations for this. The basic gist of it is, nobody knows why but everyone has a guess.

EDIT: as another user (ABCDE) correctly pointed out, keep in mind that this works differently for open-ended vs. yes/no questions.

Your second question has a general answer. Most languages use tones, which means tones change in the course of a sentence. If the tone changes for all sentences, then it also changes for questions. I know that's not what you were trying to ask, but that's the answer to the question you did ask.

If you need a way to indicate that something is a question, you could do what English does... You could use question words at the beginning of the sentence. You could change word order. You could add extra words... Which is to say, you're not dependent on intonation, though you could use it if you want to.

There are languages that use words or particles of words to indicate a question, for example Turkish and chinese languages.

Semi-related question since people have shown counterexample for OP's question: Are there English sentences where the tone goes up at the end, but is not a question? It feels like that particular tone is exclusive for questions.