Do musical numbers really happen in Disney movies?

[Hector]

Doesn't it seem a little weird when, in a Disney movie, the
characters interrupt what they do in random moments only
to start singing and dancing? Furthermore, sometimes their
choreography's synchronization is perfect as if they have
rehearsed for days! What do you think of that?
Myself, i believe what we see during songs is an audiovisual
filter, something that takes us into the characters' inner
thoughts and feelings, it is not what is really happening at
that point.
For instance, let's examine some of Lion King's songs...
-Can't wait to be king: Simba had a plan to get rid of
Zazu. Obviously, that plan could not be 'let's start singing,
hoping that every animal around will get carried away by
the rythm'. What really happened was that Simba and
Nala saw a group of animals and ran among them so that
Zazu lost sight of them.
-Be prepared: Scar called the hyenas nearby and started
whispering the plan, but what we saw instead in the filter
was him and the hyenas singing it out loud as if they wanted
the whole savvanah to know their secrets.

[Scrooge MacDuck]

That whole thing is known as the Musical World Hypothesis. Here are the main ones:

1) Could be that they're "really" singing and that's just the way this fantasy world works; it's the Musical Universe Hypothesis. Some funny examples are such that there are a few people who don't like singing but are sort of forced to go along with the flow of the universe even though they don't want to or when it's not actually to their advantage. This is cleary the case in Tangled, for example, and could be the case in a lot of Disney movies. That's just how the universe works. Very Discworld, very amusing if done right.

2) In a slightly alternate version, the "singing a speech" idea, the songs don't just burst out of nowhere — people write them down and rehearse them ahead of time — but songs are seen as a legitimate means of public communication, akin to making a speech, even in a serious context. This is probably the case for at least some of the songs in The Nightmare Before Christmas (particularly This is Halloween, The Boogie Song, and Town Hall Song). The Great Mouse Detective is also probably this, with Ratigan's song (World's Greatest Criminal Mind) being obviously real since we see him record another song later on (Goodbye So Soon) where it's narratively relevant that it's a song Ratigan composed; it's not at all hard to believe Ratigan would be vain enough to set up the whole Criminal Mind number ahead of time.

3) Yet another version uses songs to sum down what was actually spoken dialogue, as a means of storytelling, altering the words slightly; the "adaptation hypothesis". I think this could be the case in The Lion King if it's not 1. Scar went from his early whispers to a progressively louder and more manic speech, but never started singing outright. 

4) Often used in conjunction with 3, the fourth version uses songs to express fantasies and internal monologues of characters — so the events of the songs are mere fantasies and did not actually occur in any way. Also known as the "all in their head" hypothesis. (A slightly tamer version of it is when a song plays over clips of the characters interacting without singing on-screen.) The Lion King could also have an example with I Just Can't Wait To Be King, combined with 3 — Simba talked at lengths about all the things he'd do as a king, but the numerous animals comign out of nowhere as background dancers were just how he imagined it would go.