Hey Pedja,
here are my solutions. I hope i did it the right way

I am looking forward to your answer.
1) F Ionian
I IV V I: F Major, Bb Major, C Major, F Major
II V I : G minor, C Major, F Major
2) G# Dorian
I IV: A# minor, D# Major
I IImin: A# minor, C minor
3) C Phrygian
Imin bIImaj: E minor, F Major
Imin bVIImin: E minor, D minor
Imin bIImaj bVIImin Imin: E minor, F Major, D minor, E minor
4) E Lydian
I II: A Major, B Major
I VII: A Major, G# minor
I II VII I: A Major, B Major, G# minor, A Major
5) A Mixolydian
I bVII: E Major, D Major
I Vmin: E Major, B minor
6) F# Aeolian
Imin IVmin Vmin Imin: D# minor, G# minor, A# minor, D# minor
Imin bVI bVII Imin: D# minor, B Major, C# Major, D# minor