I'll add an answer just to show how using standard key signatures and chord spellings will probably make things easier to understand.
The main reason to use
D# for minor keys is the arrangement of tonics on the circle of fifths where starting with
A for minor keys ascending by fifths gives us keys with sharps and descending by fifths gives us keys with flats. The practical implications of that are:
A# minor would require 7 sharps!
Bb minor requires only 5 flats. Granted that's a lot of flats but it still is easier to read.
D# minor would use 6 sharps which is the same number of flats for
Eb minor. That would seems to make them about the same, but when raising the
^7 scale degree for the leading tone we would need a double sharp
D# minor and only a natural
Eb minor. Double sharps and flats are considered hard to read.
Also, spelling chords in thirds makes their identifies and inversions clearer. I could write
C F♭ G but it's much clear to write
C E G to see it is a triad. The other - with
F looks almost like a suspension at first glance.
The original spellings...
A# minor --> E#, A/(G##), B#
D# minor --> A#, D/(C##), E#
G# minor --> D#, G/(F##), A#
...it may seem like a subtle difference, but playing double sharps and flats is confusing, because you end up playing what seems like a natural, ie.
Gx looks like
If we use the standard key signature and spellings in thirds we get...
Bb minor --> F, A♮, C
Eb minor --> Bb, D♮, F
G# minor --> D#, Fx, A#
...again, it may not seem like a big change, but we eliminate two double sharps and an unnecessary sharp.
I know you did not ask directly about key signatures, but...
...I seem to get some weird concoctions for the indicated minor
I think the reason you got weird and confusing results isn't from misunderstanding how the dominant chord is constructed in minor keys, but by not following the conventions of key signatures and chord spellings. I hope explaining those conventions helps.