No need for such over-complicated and over-engineered solutions.
Each gun already imprints a unique microscopic signature on a bullet and casing. Just submit a scan of a fired bullet and cartridge to a central database for each new firearm sold, where it's linked to the serial number. They're test-fired before use anyhow, so I'll bet the manufacturer could easily add a forensic-type scanner to the manufacturing process, likely completely automated as well. Then we wouldn't have to rely on serial numbers on the gun at all then. Any bullet in good condition could likely be linked with the serial number using that database.
Law enforcement already uses this technique, but can only perform matching tests if the gun or additional fired rounds are found. This wouldn't require any new technology at all, only new procedures.
Not really. Those "unique" qualities are only such for a limited time.
Using, or even cleaning, the firearm will cause them to change.
Someone wanting to use the firearm, and change it's ballistics characteristics needs only a few minutes with the a few tools. File, sandpaper, perhaps a punch.
If the gun is fired and then tossed, they can match them. Fired and kept and the owner wants to change it's profile, it's really easy.
Get yourself a firearm, and use it. Stop watching so much TV.