Don't forget that the 5-lines stave is a relatively new development, I couldn't tell from the top of my head exactly when it became standard without looking up, but for centuries it was 4 lines, and previously to that it was only 1 or 2 to make those early non-dotted neumes more visual in terms of high and low.
You also know that it wasn't just bass and treble clef to accommodate music, but singers especially and some transposing instruments use a range of different clefs. It is all about the best possible arrangement of the notes available for that particular instrument (or the range of the singer), without using many ledger lines as we do nowadays on piano.
So, if you are a composer or a conductor for example you are expected to be able to read these and also to read them transposing so to insert into the whole texture of the orchestra. (Viola players use alto clef to read their music.)
Because we don't have a big range on the gurdy anyway, it uses the same treble clef. I think you should think D when you play in D and yes, adjust your fingering, don' try to transpose.
In my opinion, and it is totally personal, the D tuning must have been an answer to the widespread use of other D instruments, especially pipes and whistles in folk music. And their D-ness could be down to the size of a conveniently sized piece of wood with those holes spaced away just right comfortably? This is a guess, anybody has looked up what's the truth here?
Instruments and scales developed hand in hand:
The Medieval scores did not use accidentals except the B-flat and later the F-sharp (as if you had 3 scales only, one from C, one from F, one from G). The scales were based on tetrachords plus a tone, there is a good explanation about the basic permutations here:http://en.wikipedia.org/wiki/Tetrachord#History
The scale ideas of the Western music (and further away too) are based on the harmonics, so take the fifths from an F, F-C-G-D-A-E, organise them into steps and you have that 6 notes scale that is often used in Medieval music. The B and B flat acts as a divider to decide what mode or scale you use. This is not an accurate and detailed explanation how it really worked, but gives a good general idea.