Could it happen, that there would be C# in the key signature but no F# ? (or in general any higher # without lower one.)
It's possible, but extremely unusual. Béla Bartók was one composer who sometimes wrote things like this. For example the study no.99 in volume IV of his collection of piano studies "Mikrokosmos" he writes a key signature of just an E-flat in the right hand and a key signature of F-sharp and G-sharp in the left hand.
You can do this, but imagine yourself in the performer's seat. If you hand out sheet music whose key signature is unfamiliar it will immediately raise questions:
- Is this a misprint?
- What key am I actually in?
- Why's it been written like this?
And the answer might be "Weird! I don't trust this sheet music".
Whilst it's always been possible to read the accidentals from a regular key signature, mostly we don't; instead we see there are three sharps or whatever and know what notes to modify as we read the stave. Non-standard key signatures aren't as comfortable to work with.
You'd achieve the same sound by using a standard (non-depleted!) key signature and placing naturals in front of some of the notes.
I've seen this.
In my case, the context was playing folk music from Eastern Europe and the Middle East. Here, scales other than WWHWWWH are common, as are "interesting" compound time signatures, like 11/8 etc...
An example I remember was a piece in D minor using the harmonic scale, so the key signature had Bb and C#. It saves on accidentals, and once you become used to it is perfectly acceptable to work with.
Not if you are following standard notation for the conventional, European-based major, minor or modal key signatures. In all such scales, the sharps follow a set progression. This ensures that any note that can be notated without accidentals can be played in a scale whose notes are in the same relationship to each other as the white keys on a standard piano. In other words, the standard key signatures all define scales with the familiar whole step, whole step, half step, whole step, whole step, whole step, half step progression (although they may start at different places in that progression). There is no way to map that to a key signature with a C# and an F natural (there's too long a run of whole steps). There are some keys with a C# and an Fx (double sharp), but those are considered non-standard, and are usually mapped to equivalent scales that use flats instead. (As noted by @supercat in the comments, there is a different pattern of steps that would give you an F natural and a C# in the d harmonic or melodic minor scales, but that would not be reflected in the standard notation for the key signature, but instead, would be covered by accidentals as needed.)
If you are departing from the above assumptions, you can have any key signature you want. There are other cultures that do use different scales in "nonstandard" keys, as well as some adventurous composers who have invented their own.
It's worth noting here that the way pianos are tuned represents an almost wholly artificial ("tempered") scale designed around the (dubious) goal of making sure that every standard key signature sounds exactly the same (relatively) as every other. With "natural" tuning, this wouldn't be the case. (In natural tuning, it's also not true that B# is the same note as C or that Fx is the same note as G.)