I'm gonna give a general answer since I don't know your level of music knowledge.
The cylinder has a certain speed and length, so music must be written in a certain way that it fits there in a pleasant way if looped. This means ending in a fermata, sort of fading (and not truly looping) or having the number of bars be somewhat musical (often powers of 2).
From knowing the tempo of the song and the speed of the cylinder you know how many notes of a certain length you can fill in one rotation: 8 full notes, 16 half notes, 32 quarter notes, and so on. [1] Just divide the circumference by the numbers and you get the physical length of each note. Keep in mind that if you're not truly looping and is ending the song with each revolution, you can be very fast and loose here.
So the written music itself, plus the physical lengths of notes, will inform you on where you should place each note.
For multiple pitches: pitches are just in a different axis. One axis is timing, the other is pitch. This is visible when you look at the "comb" of music boxes, they have pitches arranged from low to high, side by side, like harp strings or piano keys.
About syncing: it is all written in sheet music, so you just follow it. Two notes that must be played together are just on the same axis. For a more tight timing just align it... It won't be perfect if you're a kid doing it by hand, which is why music boxes traditionally often use arpeggios in detriment of more blocky chords.
Multiple instruments: if a music box can play different timbres, it's triggered by having duplication, meaning two or more sound producers (combs, hammers + strings, etc). So you have, for example, 20 notes available for one instrument and 10 for another. But it is all on the same cylinder, so it is in sync. There will be repeating pitches here and there, but it depends on how the music itself was written.
The first pass of converting sheet music to “holes” must be done by someone able to read sheet music at an amateur level, and do the basic calculations (but of course some people with more experience and musical knowledge will do it better). Then they make a pattern that is reused by whoever is manually assembling.
Tolerances are larger than non-musicians expect. There is a charm to the sound of a music box, and part of that is due to minor timing issues. And that’s true for any music not performed by a computer, really. Humans are not perfect anyway, and a lot of techniques and styles rely on this (from drum flams to Dilla beats)
Not to mention that music boxes are mechanical, so it won't be perfect during playback anyway.
[1] https://en.m.wikipedia.org/wiki/Note_value