I just checked. I have three AMEX cards (Platinum, Blue Cash Preferred, Green) in my wallet. All three have both four-digit numbers on the front as well as three-digit numbers on the back.
I don't recall ever being asked for the three-digit code on the back (like I am with most VISA/MasterCard credit cards). It's always the four-digit one on the front.
The Apple Store (as in, the actual brick and mortar establishment) started using the 3-digit code on the back of my American Express card at some point instead of the 4-digit one at the front (but they are so far the only company I have personally run into which does that).
It does - there is a 3 digit number on the back in the signature strip. The Amex website will ask you for it if you attempt to change your phone number, amongst other similar high-risk transactions.
I have been asked the 3 digit code when booking an airline ticket on a non-US site. During payment it passed through a amex safekey site which I think is kind of like 3d secure