Am7 - D7sus4 D7 - Gmaj7 - E7sus4 E7 x 2
Diatonic to a key signature of one sharp,
G is a reasonable tonic. The
E7 is a secondary dominant to the
x 2 means the progression repeats. Root progression by fifths.
Am7 Bm7 - Cmaj7 D7(9) - Gmaj7
Also a tonic of
G. Note the root progression by steps, not fifths, I think that may be relevant to the "interlude."
For the next stuff, tonicize may be a better description than "change key."
Fmaj9 Fdim7 - (Interlude) Cm9...
C, notice that
Fdim7 could also be called
Bdim7 which is often theoretically explained as an incomplete dominant seventh flat ninth chord, which would be a
G7b9, the mode switches to minor,
Cm, but that doesn't matter.
...F13 - Bbm9...
Bb. In terms of keys it's "far" away from
G major, but you could think of this as just a whole step down from
Cm, sort of a sequential move.
If you accept the idea of
Bbm as temporarily tonicized, notice that the change of tonics by step reflects the root progression by step in the earlier passage.
...that chord makes sense as a subdominant of
Bb, but it also could be an inversion of
Cm which would just shift us back up to tonic
Cm. Either chord,
Cm7 can be viewed as a "borrowed" chord in
G major, which is a way to get things back to
Note that from root
Cm9 to the
Eb it's root progression by fifths. And if you regard the
Fdim7 as an incomplete
G7b9, then it's descending fifths
G C F Bb Eb.
It's important to recognize those roots by descending fifths, because it's a very important harmony concept, and the progression of roots is more important than the details of the chord qualities major/minor/etc.
...Am7 - D7...
That seems to bring us back to tonic
The part you labeled "interlude" seems like a "B" section of a song which is where you often find sequential harmony or diversions to other tonics. Because those moments are of short duration, they sound temporary, and so tonicize can be a better description that "key change."
You would need to show more detail, especially about melody and phrasing to say more re. tonicize versus key change.
...But where did Fmaj9 came from? Where does that belong to?
At least by my description, I'm saying it comes from the tonic
C. It's the subdominant of
C, and it was arrived at directly, there was no pivot chord, or that kind of thing. Moving from an initial tonic of
C is perfectly ordinary.
If there is confusion about an
F major chord in relation to a
C minor tonic, and the changing between the tones
A natural and
A flat, understand that this would all be described as modal interchange or borrowed harmony. The critical idea is chord roots and tonal scale degrees are the principle elements of tonality whereas modal chord and scale degrees are only elements of modal coloring.