Ianober,
If the case was metal you would usualy put a lug on the case and the bare ground wire would be attached to it. That would protect the user should he touch a hot box ( the circuit breaker would pop if there was a short to ground ). If GFI protected then if a small current was sensed the breaker would pop. The transformer is usualy bolted to the metal case so you have a path to the earth ground. The transformer may not have a ground wire, the green with yellow strip could be a tap for a lower voltage off the secondary windings of the transformer. The primary and secondary windings of the transformer are suported by the transformer frame so attaching the ground wire to one of the mounting bolts of the transfomer would be acceptable because to you are using a plastic box. Just remember that i am not an electrician and safety is your sole responsibility!
Let me know how you make out with the cable check.
FWIW.....I have been were you are.......so smile as today is just another learning day!
RICH