Here is what I know about NICS. I worked a part-time gig at a Sportsman's Warehouse Gun Dept. for a couple of years. There are two NICS call-receiving centers. No telling which one you get when you call. They simply enter full name and DOB of buyer, along with SSN if given. IF no criminal record is found, they give the go-ahead.
IF a near-miss to someone with a criminal record comes up ( maybe different middle name, transposed DOB, slightly different spelling of last, etc ) they transfer to an actual FBI data base, who continues the check. That should clear up near-misses. IF the person being checked has a criminal record, that may or may not preclude them from purchasing a weapon, they refer it to another person, which means either they phone back, or just not phone and let the time lapse if the person is cleared to buy.
Each state is different regarding CCW's. Here in Washington State if a person has a CCW and buys any weapon, the NICS is still done. BUT, with a CCW a long-gun can go out the door as soon as NICs clears. With a handgun, even with a clear NICS, there is a waiting period before it goes out the door ( it's been a few years, I forget how many days, 3 or 7 or some such ).
A person who is often or always delayed can get a packet from most gunshops. It will have a fingerprint card and other stuff. Follow the directions, submit to address provided, and you will be given a "code". Next time you buy a weapon, provide that Code # to the clerk when they run the NICS check, so they can give it to the NICS call-receiver. That should provide an instant approval.
In my case, about 20 years ago I plead guilty to a gross misdemeanor. This offense does not preclude me from buying/posesssing firearms, but I am always being delayed. About six months ago I went to court to have my conviction "vacated". A criminal check on me now shows no convictions, but I am still being delayed ( yes, I have had a CCW all along). So now my next step will be the packet that I mentioned above, to keep from being delayed.
It's a bit of a hassle, but really, compared to other countries, it is not too bad. Mark