Many good points have been made in other answers and comments, but I think we need to unpack some of the assumptions (implicit and explicit) made in your question.
ASSUMPTION 1: The world's languages can be divided into the binary categories of "tone languages" and "intonational languages". This is false. All languages have intonation. A subset of those languages employ lexical tone--defined broadly as the use of pitch at an abstract level to encode lexical contrasts (i.e., meanings of words).
ASSUMPTION 2: Pitch is not used in non-tone languages to distinguish word meaning. Also false. Languages like English, which has lexical stress, use pitch acoustically to signal stress, which in turn can distinguish otherwise homophonous words with different meanings. For example, ivy vs. I.V. (intravenous device).
ASSUMPTION 3: Pitch is the only thing that encodes utterance type (like question vs. statement) in non-tone languages like English. False. Most questions in English are actually NOT cued by pitch. Wh-questions--those starting with words like who, what, when, how, etc., generally have an intonation that is similar to declarative statements. In such questions, it is the presence of the wh-word at the beginning and the inversion of the word order that signals the question. Yes-no questions most often also include changes like the inclusion of DO and the inflection for tense on DO ("Did you go to the store"), and while their intonation is often rising and distinct from that of statements, it doesn't have to be--I can say "Did you go to the store" with a pitch contour that goes down at the end if I'm being somewhat insistent and confrontational. The only type of question that is consistently cued solely by pitch is the intonational question--"You went to the store?"--which can be echoing a statement someone else just made or expressing disbelief or surprise.
ASSUMPTION 4: Pitch can only cue one thing in an utterance at a time. Again, false. Even in non-tone languages like English, pitch encodes many things in an utterance simultaneously. For example, take the utterance:
You returned Maria's IVY to the store?
Kind of a weird utterance, but the pitch contour on the word ivy is simultaneously encoding three things. One, it's signaling that the stress of the word is on the first syllable, thus distinguishing it from the word I.V. and effectively encoding lexical information. Two, it's signaling that the sentence is a question (if it were a statement, we'd get a high pitch point on the "I", but instead we get a low pitch point there because it's a question). Three, it's putting focus on the word ivy, letting the listener know that the speaker is asking about the item that was returned and not, say, where it was returned or whose item it was.
Now, all that said, the question remains--if all languages have intonation, and if some of those languages also have lexical tone, then how is pitch used in those languages with lexical tone to encode intonational functions? That is a good question, and not one that can be answered with any generalization--in other words, it depends on the language. My response to ASSUMPTION 4 above gives you some clues as to what some of the logical possibilities are, and @user6726 provides some good points and examples in this regard. Different languages reconcile tonal and intonational pitch cues differently. Please see my responses to How do sentence intonation and (syllable-based) tone interact in tone languages? and Are there tonal languages which use a rising intonation for questions?, respectively, for more on that issue.
As for your questions about singing in tone languages, that's been covered elsewhere, as @fdb and I note in the comments above. But here's a hint: if lexical meaning hinged entirely on pitch in tone languages, how could speakers of those language ever communicate with whispered speech?