It all depends...
Sometimes, the job will spec a seperate ground wire. By code, EMT is permitted as the grounding path, and a ground wire does not have to be run in EMT. (Sorry, i dont have the book in front of me, so i cant tell you the section.) Usually, a seperate ground wire is run for an added measure of safety.
BX is rarely used anymore, and it does not contain a ground wire anyway.
A receptacle must always be bonded to a metal box, unless it is a self-grounding receptacle. if a ground wire is run, it must be bonded to the box and the receptacle, again, unless it is a self-grounding recetacle. (the ground wire still must be bonded to the box in this case)
Wire gauges are as follows:
The grounding jumper must always be the same size as the circuit conductors.
I don't believe it is allowed to recode a white wire green the way you described. if you are using AC (BX), the metal sheath serves as the ground, so I don't see why you would need to use the white wire as the ground anyway.
All you would have to do is bond the receptacle to the box, and it would be grounded.