This looks like a classic case of old technology being overlooked – I managed to dig out a copy of Graphics Gems IV from the garage and it looks like Ken Shoemake has not only an algorithm for converting from Euler angles of arbitrary rotation order, but also answers most of my other questions on the subject. Hooray for books. If only I could vote up Mr. Shoemake’s answer and reward him with reputation points.
I guess a recommendation that anybody working with Euler angles should get a copy of Graphics Gems IV from their local library and read the section starting page 222 will have to do. It has to be the clearest and most concise explanation of the problem I have read yet.
Here’s a useful link I have found since – http://www.cgafaq.info/wiki/Euler_angles_from_matrix – This follows the same system as Shoemake; the 24 different permutations of rotation order are encoded as four separate parameters – inner axis, parity, repetition and frame – which then allows you to reduce the algorithm from 24 cases to 2. Could be a useful wiki in general – I hadn’t come across it before.
To old link provided seems to be broken here is another copy of “Computing Euler angles from a rotation matrix
“.