Are you in the US? How did you get a Euro Z3 here?
The 97 model year did not have factory keyless entry. It only had an accessory Alpine alarm/keyless kit. Does your car have this? You may have to remove the glove box and look for the alarm unit. That unit uses an external alarm fob. There are no buttons on the key. The 01 and 02 Z3s (I think) did have keys with buttons, but yours does not. I don't know what key they sold you or why they did it.
Rear fogs are great. When it is foggy, turn it on and drivers behind you will see you in the fog much easier.