This is best observed in SATB choir music actually.
When the music is written in short score (e.g. SA on the top treble clef stave and TB on the bottom Bass clef stave) then often you find that although the tenors are singing well in their range on say a G (392 Hz) above middle C then in this short score they are singing a note which is 3 ledger lines above the top of the bass clef which, as a tenor, is extremely annoying to read when you first start singing in bass clef but after a couple of months it becomes second nature.
When music is written in open score or full score then each part S A T B has its own set of ledger lines and the tenors sing in treble clef -8ve which means they sing everything down an octave from the written pitch and means that the same note G (392 Hz) sits neatly on top of the stave.
This is the same for every other instrument. Some instruments don't fit onto either the treble clef or bass clef very easily and so they use a different clef e.g. violas read in alto clef.
Another reason for instruments being in different keys is not only so they fit into different clefs but is also due to their construction. A tube in Bb is in Bb because it's the lowest note it can play (with out pedalling down) and that has an effect on what notes it plays on an open fingering and what notes it has to use valves for (i.e. notes that do and don't fit into the instruments harmonic series).
I may have missed some thing out very obvious here but that is my contribution to this answer and I hope it helps.
I would argue that actually C probably isn't as big a centre of music as you might think! You probably only think it because in it's major key it has no incidentals at all but if you ever hear an orchestra tune up they tune to what is called 'concert pitch' which is A above middle C and defined at 440 Hz. I would argue that A is the centre and the only reason is because it is the first letter in the alphabet (note that full size grand pianos with 88 keys also start on A and that it is only MIDI controllers that start with C because for some reason they go up in banks of Octaves starting on C).
And lastly to you statement about D being the historical centre of music because of its link to the Dorian mode. Actually the dorian mode has nothing to do with the note 'D' other than it's intervals happen to sit perfectly with no black notes on a piano when you start on D. Historically, and especially in the church where the dorian mode was arguably used most, they wouldn't pitch a D and always sing with the root note on a D when singing in the dorian mode. They would just pick a random note and start from there but they would get the intervals correct which are this starting from the root, tone, semi tone, tone, tone, tone, semi tone, tone. It really has nothing to do with the note D. You can sing the dorian mode starting on any pitch...
Hope this helps