PG,
Q1. You will not be expected to remember such a complicated formula in a test. You will be expected to know and compute with just the basics, and will be given the obscure background equations if needed. You will not find the same problems involving time consuming proofs contained in an exercise book on limited time test.
Q2. This has nothing to do with the rules of vector manipulation. It is an application of ordinary algebra. The last term D(A.BxC) is first equated to the other three terms. Then the equations is divided by A.BxC resulting in the next equation D=(three terms).
Q3. In the numerator of the second term B(A.CxD) = B(AxC.D)=-B(CxA.D). So reversing the terms C and A of the cross product causes a minus sign to appear, which cancels the minus sign present in front of the second term.
Ratch