No, the Troy Safety Selector Assembly will not work with the Franklin Binary System. From what information I have read the Franklin Binary Triggers have to be used with the Franklin Armory Selector compatible with the model trigger you are using. I recommend contacting Franklin Armory in regards to a replacement selector.
Yes, it will work with your AR-10 build.
Yes, it will work with your .308 build.
Yes this should work with a M&P 15 scout 2