I interpreted the intro Em/C# (from c# Locrian) | D#m/C# (c# Major) | D/C# (c# Phrygian) | C#m (Aeolian) as Im | % | % | % | with different polychords on top of c#, written in different modes.
This passage is minor triads chromatically planed against a C# pedal tone. Contrapuntally, the harmony unfolds the third between E and C#, which are both in the tonic triad. You'll drive yourself nuts if you try to justify everything with some one-chord modal nonsense like you're doing here. You are correct that the passage prolongs the tonic chord of C# minor. Chordal analysis does not really work here, to be honest.
So, to me the composer is just playing with modes over the same c# note.
You ignore that Gore is using the specific color of the minor triad. This rather blatant feature (which is exploited throughout the song) is minimized when you start saying that each of these chords represents a different thing. Look for unity, not disunity.
Then we have: C#m (fixed) | G#m/C# | A/C# | C#m | E#m | A (Lydian) | F#m (Mixolydian) which I think is, once again, Im | % | % | % | IIIm | bVI | IVm
Yes, but once again I'd ditch the modal stuff. It's i [v ♭VI i] ♭VI iv with a chromatic mediant thrown in (i [v ♭VI i] iii ♭VI iv). Nothing beyond the pale of traditional tonal analysis. Stuff in square brackets is prolongational.
where E#m is borrowed from the c# major scale.
This is one way to think about it. These third relations are all over Depeche Mode's music, and I would aim toward generalization. The important thing is the chromatic mediant relationship. I don't think we hear it as mode mixture (as we would if the harmony went from C#m to C#, or C#m to F#).
In this case, the chord progression seems to be quite typical, with IIIm extending the tonic Im chord.
You are absolutely correct: iii prolongs i in this context. (That may not always be the case, but in this song it does.)
In the final part there are Em/C# (Locrian) | D#m/C# (Major) | D/B (B Aeolian) | C#m that should be simply Im | % | bVII | Im quite typical. Can you please check if my interpretation is correct or if you have a different idea?
It's the same as before: chromatically planed minor chords unfolding the root and third of the tonic chord. The last one in the sequence, Dm/B (the score and your analysis is incorrect), creates an interesting sonority: Bø7, or ♭viiø7. This is a case where chordal analysis short changes us. Contrapuntally, the outer voices are approaching the tonic pitch by contrary motion: ♭7-1 in the bass, and ♭2-1 in the soprano. I would focus on ♭7 acting as a leading tone or implying ♭VII. (Tricky, because "leading tone" would traditionally mean B#. However, in context, its harmonic function as a tone in a pre-tonic chord is plainly obvious.)