Semi-noob questions - Grey market? Used etc.

[Doombot!]

Can someone please detail what the Grey Market is? I know what an AD is, but how do they differ?

Also, will/can an AD verify a used rolex is authentic (if so, how much would be a good price to do so, and how long would it take?)

I'm hoping to be in the market for a SS sub and it seems many hundereds of dollars can be saved on the 'bay; but I'm scared.

Finally, why is there a big price break between the date vs. no date sub's?

Thank you!