Yes, you can look in BSI and look for fob information.
The starting and remote are 2 different parts of the same key, so its easily possible to have a C3 key that starts but won't unlock or unlocks but won't start.
You should get the key card when you buy the car, same as a spare wheel, jack, spare key and owners manual. I don't think a C3 could ever be cheap enough without all the parts considering how many of them are out there for sale.
It is part of the COM2000 unit.
There is The COM2000 (or steering wheel stalks) - an inside look - or teardown topic for reading.
No dedicated fuse.