You only need to show your pass when you enter the park. Once you get in, no need to go back to the entrance.
The discount at the campgrounds applies to NP operated campgrounds. It usually does not apply to concessionaire run campgrounds though some will honor it.
For those not old enough to get the SR pass, recommend the Annual Pass. Makes it so easy to go into a park and usually pays for itself after two or three parks. This is what I use and will for the next 11-1/4 years.
I have never been given a receipt when accessed with my annual pass and would suspect the same for the SR pass.