All of the modes — Major, Dorian, Phrygian, etc. — can start on any key. Using major as the starting point, then the "first" mode is major, the "second" mode is Dorian, the "third" mode is Phrygian, etc. Modes use a common key signature but differing tonic pitches.
For example, if I begin with C# major, then I will have, corresponding to that key signature, D# dorian, E# phrygian, and so forth. From D major, I have E dorian, F# phrygian, ....
The sharps and flats are no exception. Any scale can start on a black key or a white key.
If one plays only the black keys, as expressed in the question (
C# D# F# G# A#), the result is known as a pentatonic scale: specifically, the third mode of the F# major pentatonic scale. Pentatonic scales, as any other scale, may start on any key, black or white. A C major pentatonic would be
C D E G A, C# major pentatonic would be
C# D# E# G# A#, D major pentatonic is
D E F# A B.
Search this site for "pentatonic" for many posts on that topic.