I am trying to transpose the key signature up a perfect 5th from C minor, but am unsure what the new KS would be. The book only tell you to write the new tonic and KS without explaining how the key changes. This lead me to thinking the new key was in G major counting 5 from C (like intervals) or A Major; excluding counting the tonic? - I am probably missing something very simple here, any ideas?
If you know enough to transpose the actual music, then you know enough to transpose the key signature: You knew that you transposed everything up by a perfect 5th, so transpose the key of C minor (three flats) up by a perfect 5th, resulting in the key of G minor (two flats).
This means counting the tonic as one, which is easy enough to conceptualise: There's no way to transpose by a "zeroth", so not moving at all is the "first" (unison).
We don't do your homework for you here. But here are some hints:
If you transpose a tune that is in a minor key, it will end up still in a minor key.
If we're using key signatures, accidentals will be required on the same notes before and after transposition. Look at your Question 4. The 7th and 13th notes in the tune (I'm talking a simple count here, not intervals) didn't fit the key signature, so need accidentals. The 7th and 13th notes of the transposed version will similarly require modification.
When transposing, it doesn't really matter whether the tune is in a major or minor key. The transposed notes will be the same. Consider a very simple 3-note tune G, A, B. Key signature one sharp. It's to be transposed up a minor 3rd. Now, it might be in G major. Up a minor 3rd from G major is B♭ major, so write a key signature of two flats. There are no accidentals, so the transposed version will be B♭, C, D.
Or perhaps it's in E minor. No way of telling. But no matter. Up a minor 3rd from E is G. G minor has two flats, so write that key signature. The transposed notes will still be B♭, C, D!
(Check your answer to the first question. What's the key signature of C minor? Which note is changed from the key signature in a Harmonic Minor scale? What do you get when you 'sharpen' a note that was originally a flat? And it asks for crotchets.)
(And someone tell the setter that the orchestral convention is that French Horns don't use key signatures. Difficult to argue why, the traditional reason is well obsolete. But it is so.)
For the specified transcription 'up a P5', you either add a sharp, if there are any in the original key, or take away a flat, if there are any in the original key.
Transcribing a minor will be minor in any new key; majors remain majors.
If you're asking about any transcription to any key, it's a long answer, suffice to say use the original root note (with its related key sig.), move it up/down to where it needs to go, find the new key sig.from the new root note. Most players will end up learning the different key sigs. There are twelve, including the relative minors, and knowing the circle of 4ths/5ths will facilitate this.