Basic transcription and analysis questions are supposed to be off topic.
But, part of the confusion is the software. The chords are not simply
C#m ii V i IV
I'm not going to transcribe and analyze the whole thing, the problems can be understood by the first chord alone.
First, it is not
C#m: ii. If we keep it to just a simple seventh chord label, without putting in non-chord tone stuff, it's a half diminished seventh chord which is
C#m: iiø7 or in jazz symbols
D#m7♭5. If you try using the software progression label, it's confusing, because the actual part isn't simple triads. The
V also looks like it should be an altered chord, some kind of
Next, the full tones of the chord ascending are
D# A C# F# G#, reorder that as a tertian stack to get
D# F# A C# G#. That's a
D# half diminished with the
G# as an eleventh, extended harmony or
sus/add doesn't really matter at this point. The important thing is we have a clear root position half diminished chord.
You could reorder the tertian stack using a different supposed root:
A C# G# D# F# and call it an
Amaj13 with the
...I don't understand why the bass note is only used as a bass note for another chord (e.g., Amaj13 / D#). What theory describes this?
Analysis is what describes it. A proper analysis would first determine what the tonic is, but we can just stick with the assumption
C# is the tonic and the mode is minor. A bass movement of
D# G# C#... - descending perfect fifths - certainly reinforces
The question then is why
D#m7♭5 is not simply interchangeable with
The answer won't come from just looking up a list of tones in an online tool. You need to understand principles of chord voicing, especially how inversions and the bass work.
For basic, practical harmony, you normally have the chord root in the bass with movement by fifths/fourths. When the chord is inverted other chord tones like the third, fifth, seventh are in the bass, and the bass will move step-wise. Additionally, with extended chords the extensions ninth, eleventh, thirteenth are not normally in the bass.
So, we have the choices of:
D# bass for a root position chord, moving by descending fifth to
D# bass for an inverted, extended chord, with a sharp eleventh in the bass, moving not by step, but by leap to
Basic voicing and voice leading principles say the first options is typical.
It's much more straight forward to say the
D# bass is a root to a root position chord than the eleventh of odd inversion of an extended chord.
Follow that principle for the other chord, which are in root position, and then go back and fix the Roman numeral analysis errors of the software label for the progression, and things will make more sense.