From my experience with the plastic cones, they need to be washed daily with antibacterial soap, the green color you are describing I think is the minerals in the water in combination of the bacteria and slobber from drinking, mixed with the plastic. This happened to me from not washing the cone for a couple of days and my boy has an extremely wrinkly 'turkey' neck, so the water from drinking got trapped in a pocket created by the cone closing up two wrinkles.
If it were me, I would remove the cone, clean it really good, and clean the neck area really good, try to keep it off if you can 100% supervise, like a good cuddle on the couch, only put it on when you cannot supervise.
I agree with above posters on going back to the vet to see if there has been any progress. Certain eye meds can make a scratch worse and not better, please let me know what eye med you are currently using. If it has any ingredients with a 'zone' at the end, (like hydrocortiZONE, predniZONE, ect) then you need to go see a new vet right away!!! Fire your other one, for sure!!!