- What is this a#m7(#5) chord? What's the relationship with the chords being played before and after? Even if I see this chord being an inversion of the F#maj chord, I can't really see where this chord comes from... and goes to.
Seems like you've got a solid grip on the first few bars of the section, so I'll skip over that. And yes, I'd say E major is the best choice of key by far here. That's a good start! So what is A#m7#5 doing in E major? Well, first of all, I would point out that it is rare for a minor chord to be played with its fifth raised, since that modification ends up making the chord enharmonic to an inversion of a major triad, and so if that sonority arises, I think the most likely scenario is that the chord is an inversion of a major chord. In other words, I would steer clear of m#5 as a label unless it's really the only sensible chord symbol.
So let's ignore the symbols and just try to put a finger on what this chord is even doing here. For starters, one of the most obvious things to notice is the A# in the bass, which isn't in the key of E at all. What's that doing there? Well, it's not part of any chord in E, but it does look and sound a lot like it wants to go towards the note B (yes, that means A# is the correct spelling rather than Bb).
It doesn't go there, but let's pretend that it does for a second. Typically, this A# in the bass is either the third of an F# chord (V/V, a secondary dominant) or the root of a diminished chord (viidim/V). These chords are predominant chords that set up a move to the dominant, and they move the bass up by half-step to get to the dominant note. This is even supported by the melody, which runs up to an F# as though landing on the supertonic chord. So that's one way this chord could be rationalized in the key of E - it's some kind of secondary chord pointing to the dominant.
However, clearly here we don't end up on the dominant at all; instead we arrive at Am6 (actually, upon close listening there's no E note in the chord, only an Eb/D#, so it's really a diminished seventh chord instead, but that doesn't end up changing its function in this case). In this case, you could look at the A# chord as a substitution for the IV chord, Amaj7. In fact, the only difference between A# half-diminished 7 and A major 7 is the note A being sharped. If you replace the A# chord with an A major chord, the progression makes sense as a minor plagal cadence that ends up going to the inverted I chord.
This is a common chord substitution in jazz harmonies, and that kind of city pop uses those kinds of progressions often (footnote 1). Or alternatively, this is also similar to the IV V iii vi progression that a lot of Japanese popular music uses, just change the IV and the V is replaced by any chord that has the same function of going to iii (in this case, iii is also replaced by I in first inversion). However you slice it, the idea is that the A# chord is really a substitution for the normal IV chord. This is the same kind of chord progression that "Christmastime is Here" by Vince Guaraldi uses, if you're looking for another example in context. So this is still a predominant chord, it's just in the context of a minor plagal cadence instead of a normal authentic cadence.
So now we have a good idea of what this chord is doing harmonically in the chord sequence. I'd say now that it's best written as either A# half-diminished 7th add flat 13th (which makes the function a bit more obvious but is a bit clunky - just stick to A# half-dim 7 for clarity), or as F#9/A# (much cleaner, but is it really an F# chord?).
- I believe the 'I' [E/G#] acts like a dominant for the next chord that is 'iv#7' [amM7], but where does this chord come from?
Well, if we're considering the above chords, it made the most sense for E to be the I chord, so perhaps instead it's better to recognize that A is the IV of E. At least that way, we have another common minor plagal sound which would fit right into the music so far. The only issue I have with that is that it's a weird place to put Am(maj7), it doesn't feel very resolved going to the next chord. It doesn't continue the descending bassline that the previous three chords established, either.
I had to go and listen to the song from the beginning to get a better picture of what the structure of the song was, and I realized that the A# chord starts a new prechorus section. That immediately tells me that this Am(maj7) chord is in the middle of a section, so it makes more sense in context to play such a restless chord there to keep momentum. By the way, I don't think this song EVER lands on the tonic chord in root position, and that would make a jarring iv(maj7) chord all the more appropriate to avoid any strong cadential resolutions.
So my theory on the Am(maj7) chord is that it's simply an interesting chord to get away from the I in a semi-logical way while also not committing hard to a resolution. Notice that it is followed up by a C#m7 chord in third inversion (the Am6 in the question's chord chart is completely incorrect on that one) which immediately begins another chromatic walkdown - that's technically a tonic function chord like E6 would be, but this song really doesn't want the audience to get comfortable on the I chord. For another example, "God Only Knows" by the Beach Boys does some similar things with inversions and avoiding the I to avoid strong harmonic resolutions.
[...] another movement I don't understand comes here: from the am6 we go to a#m7(b5), which I believe is just an F#9 without the F# in the bass, because right next to it we have a complete F#9, being a secondary dominant to the last chord B6/9sus4.
The chord chart in question is incorrect again; after that A# chord is an F#m9 chord instead, just part of a regular ii V progression. So this makes a bit more sense; now the descending bassline idea that started with the C#m7/B is aimed squarely at the note A. True to form, the song jumps down to F#m instead, but that's a simple substitution for the subdominant chord that the audience expects. This progression wouldn't work as nicely if it were a secondary dominant there, in my opinion, since that A# half diminished chord needs to go somewhere and F#7 is just too similar to feel like motion.
B6/9(sus4) is the big honkin' V chord, obviously, and note that it retreats back down onto the IV every time instead of going to the I. No strong resolutions here, which is clearly a stylistic choice.
- Now, I said "I don't understand", because I was wondering... Does every chord need preparation? Like, can we throw a secondary dominant anywhere (just like this song, iv > V/V > V > IV > I), or am I missing something?
That will depend heavily on context, genre, and a lot of other factors. It's generally true that for most chords in any functional harmony setting the placement matters a bit - I iii ii iii V is not a very convincing chord progression, for example. I wouldn't say it's necessary to have a complete reasoning behind every single chord in every song, but more often than not there are useful patterns to observe and analyze in when and where different chords appear in music.
And it never hurts to double-check the lead sheet before analyzing, there were some errors that could trip people up pretty easily in this question. You can save yourself a headache or two when playing it, and analysis will probably make more sense as well if you've got the right chords!
Tatsuro Yamashita has even cited jazz, big band, and soul as influences, and the same kind of progressions can be found in the music of his wife Mariya Takeuchi.
There's a whole thing about negative harmony that basically makes the case that the minor plagal cadence is of equal strength to the authentic cadence because perfect fifths invert to perfect fourths and some crazy stuff like that, if you're looking for some extra rabbit-hole spelunking research.