The other upvoted answers here are good and I don't want to repeat them, but I think that there is a lot to add.
Surely the key of C-Sharp Major must exist! So why is it never or rarely used?
There are several factors that need to be taken into consideration. Here I list a few:
To understand modern scales you need to look at how they originated. This can be fairly technical so I will give the short version. Before the major/minor system there were hexachords - scales built from 6 pitches with 1 tone steps except for a semi-tone in the middle, 3-4. In the Middle Ages, music used 3 hexachords:
- hexachordum naturale: C D E F G A (semitone in E-F)
- hexachordum durum: G, A, B♮, C, D, E (semitone in B♮-C)
- hexachordum molle: F, G, A, Bb, C, D (semitone in A-Bb)
In the 15th century or so more hexachords were added using the same process which results in sharps and flats being added one by one. Eventually, you could have one that looks like this: C#, D#, E#, F#, G#, A# (semitone in E#-F#). It's a bit like the circle of fifths in equal temperament where you add a sharp (flat) for each modulation up (down) a fifth.
Note that in these eras there is no equal temperament yet, so C# and Db hexachords (or even major scales) do not sound the same. In general, I will consider equal temperament when answering your question.
The point is that not all scales were treated equally. Those with less accidentals had historical preference and it took time until those with more accidentals were "discovered" and used. And we all know that old standards stick and have heard at some point the answer: "because of historical reasons".
"Color" and meaning of a scale
This is a bit more fuzzy. There were (are?) notions of different "colors" and meanings for scales.
You might have heard things like "F major is very bright [compared to, say, Bb major]" or "C# minor is very dark [compared to A minor]". It could be that this was the acoustic result of the tuning system and instrument building at the time (remember that the instruments evolved with the expansion of hexachords, so before the 14th century instruments didn't need/have a D# pitch, for example).
Sometimes different scales mean or symbolize something for composers. You sometimes hear "Beethoven chose
insert scale here because it was associated with freedom [or something]". For example, Beethoven saw C minor as "powerful and emotionally stormy".
It will require digging to figure out what each scale was associated with at each musical era or for different composers. I have yet to see any such association/symbolism for C# major, nor a "color".
The scale in which we hear
Again, this topic is a bit fuzzy, but I will make my point clear. Most (all?) of us, when we compose music in our heads, do it in one of just a few scales. Try to compose a few tunes in major and minor on different days (don't listen to anything tonal beforehand) and be faithful to the key you hear them in. You "should" find out that you don't hear the music in your head in 10 different scales - you stick to a few.
I'm not sure what kind of bias it has or what the statistics are. I'm just noting that, again, not all scales are treated equally.
Ease of use
Scales with less accidentals are easier to read and play. It's also easier to modulate around them. Just as an example from A-sharp minor: "A♯ major, usually replaced by B♭ major, since A-sharp major's three double-sharps make it impractical to use." I wouldn't take Wikipedia as a gospel, but "impractical to use" is somewhere in the right direction. Nevertheless, Chopin modulates to A# major in one of his pieces.
Both C# and Db major have a lot of accidentals.
Any combination of the above can give you the answer. There are certainly many instances where those keys are used, either as the base of the piece or in a modulation, but they are truly less abundant than the scales we usually see (hear). Just remember that C# major is not the scale with most sharps as you can always take enahrmonic scales like A# major and these are used even less.
Are there situations where the key of C-Sharp Major would be more appropriate than say the key of D Flat Major?
While the previous question would be more in the area of musicology, this one is much more technical and one has to look at theory. I won't go deep, but the answer is "yes". Here is a simple example:
You write a piece in Binary form in F# major. You modulate to the fifth, which is C# (not Db) major. You then modulate back to F# major and finish the piece. This is a regular I-V-I form.