First - in the few jurisdictions that I am familiar with, and especially Texas, the one I am VERY familiar with, police officers do not confiscate guns and check serial numbers as a routine matter. 95% of the time they don't even want to see the gun, never mind run a serial number check on it.
Second - you just can't run a serial number check on a firearm. If it wasn't used in the commission of a crime, or apparently stolen, or in the hands of a known felon, nobody runs serial numbers willy nilly because you were stopped for speeding, or third degree lurking. It's just not done.
Third -
I have my CCP ,but most often carry an un-documented firearm. What are the implications or problems associated with that?
We like to call it "without administrative encumbrance". What that ordinarily means is that you obtained the firearm without a Form 4473. In most states that is still a way you can acquire a firearm legally. In some states it could mean an unregistered firearm but if you live in a state where all firearms are registered and you're carrying an unregistered handgun - and an LEO actually does check - you're going to jail. In most states an LEO cannot check for registration because most states don't have such a thing. 4473s are not registration. They're just required Federal paperwork required to be kept by a federally licensed dealer, along with a bound book record of all guns in and out of the dealership. They are the dealers' records. They are not Federal records until and unless the dealer closes up shop and sends them to the BATFE, as required if he closes his doors. I still await proof that the BATFE has some little gnome somewhere logging the data into computers. The records are stores, sure, but uploaded? Someone has yet to prove that to me and I have asked here and there many times. So, for most of us, there is no "implication" of any kind.
Just wondering if I happen to be stopped and checked. Will the officer do a serial number check on the gun and what happens if he does and finds its not registered to me?
HE WILL NOT! Well, unless that's required where you live. Otherwise the officer is unlikely to want to even see your gun, as noted above, never mind run a serial number. None of my guns are "registered to me". Even if I drive into a state where there is gun registration, if my CHL is accepted there, then what? I possess an unregistered gun in the State of Confusion - what are they going to do about it? Nothing. That's why they won't check.
An opposite example - suppose you're in New York City - it could happen; I travel there 4 or more times annually. I don't have permission to carry a gun there so I do not. Too risky! Mandatory 1 year in jail. But suppose you live there - LEO stops you and you do have a gun and you do have a carry permit - well, you better have that little piece of paper or plastic with you because in NYC he will ask you for it - because they like nothing better than an easy weapons collar. That gun IS registered to you - it's ON your permit. You better not have one that is "undocumented".
See the difference? There, they don't have undocumented guns. Having one is a crime. Most places all guns are undocumented, not counting the 4473s and bound books.
I could go on but I think the point is clear. If you do not have registration in your state all of your guns are "undocumented" as far as the state is concerned.