My local shop ALWAYS asks for my C-Card..... Just understand that the "C" stands for Credit!
As to the issue of providing a dive log to verify experience, I can only recall being asked to provide that info twice. Once for my IDC and then again for my IE. Even then, I didn't have the log so just had to make one up as best as my memory would serve....
I started diving in 1970 and going to the shop (Steeles) in Oakland CA with my dad. It was like going to a good friends house... or a barber shop where they just get to know you... Nobody ever asked anything about certifications or logged dives... I guess when they see you schlepping a dozen steel 72's from the jeep into the store.....they just figure that you're OK..
I now have a "new" documentation issue in that I'm headed on chartered Catamaran trip to the BVI in 5 weeks and since I'm over 65, they want a signed "permission slip" from a doctor saying that I'm good to go. What doctor in their right mind would write that note? Even if all they wanted was for him to say something more benign like.... "I see no reason why this person cannot dive"...... I still would not sign that if I was a doctor..