Although I'm not really a DM, if I were to DM, my answers would be:
1. I prefer a degree of explanation, but it's fine if we have to hash it out together. E.g. a Fighter who takes some Wizard levels should have found, or experienced, something to have that happen, but I'm open to a wide variety of interpretations.
2. No. Because that's a lame copout answer. "Oh sure you can become new things...by abandoning the character for this campaign!" Since I'm not saying "no" to multiclassing in the first place, making it take "years" to train in a new class would be just plain rude, a bait-and-switch if ever there was one.
3. Some classes require more explanation than others, I'll grant you that, but I don't think any class should be off-limits if à la carte multiclassing is supposed to be a thing. Druid: the primal spirits of the world call out to me in my dreams, but I never understood them until I started adventuring through the wilds. Cleric can be justified just as easily as Paladin, especially since 5e Paladins are more about Oaths now than about specifically serving a deity (I presume the new Oath of the Crown is a good example there--an oath to a monarch/y, which doesn't have to be religious at all). Ranger can take a bit from the Druid explanation, or it can be an outgrowth of having had to rough it through the wilds during prior adventures; you pick up a thing or two if you're really paying attention.
Bard is even easier, since it's literally explained as tapping into "words of power" type magic--clearly, the character has discovered or figured out one of these words of power, and in so doing accessed something special. If notified in advance, a good DM could easily pave the way for such a thing to happen a few sessions ahead of the actual level-up.
Monk...I'll admit this one I'm having a bit more trouble than the rest, in part because I am fantastically tired. However, surely a class so thoroughly tied to achieving "enlightenment" could be justified by a transcendent experience of some kind? The opening of the way: but after that moment of oneness, it takes much effort to make such a shining moment extend into every waking moment.
So yeah. I don't see any problem with any particular picked-up class, even if there is no trainer/mentor available. These magics and forces permeate the world; as likely as not, they'll find you before you find them.