I work for a Scandinavian company. We don't have any rules governing those details.
However, on the ground when the paperwork is on your lap anyway, I don't know anyone who doesn't write down SID clearances and so on and other instructions (taxi, routing, levels and headings) are to be repeated by the handling pilot after being read back to ATC by the NFP.
Regarding frequency changes, it is a good technique that the frequency is first read back after having been set in the display. It is often the handling pilot who sets this, or whoever's hand comes first, but the frequency which is read back is the frequency which is set.
One of my pet hates is a taxi clearance all the way to the runway but then having to hold before half way. "Taxi A, B, C, D, hold before C."
In these cases I only write and read back as far as we are actually cleared. "Taxi A, B, hold before C."