I'm jealous, I have 8 M1 Carbines but still don't have a authentic paratrooper model. I wouldn't do a thing to it other than keeping it clean and lubed. You can find replacement barrel bands but what you have is more valuable to a collector even if it is broken.
The carved "Jimmy" does detract from the value of your M1 unless you can find out who the person is and his story. Knowing Jimmy's background would make the weapon quite valuable (it is already pretty valuable judging from recent auctions)
...probably the greatest concentration of talent and genius in the white house except for perhaps those times when Thomas Jefferson ate alone.
John F. Kennedy, Describing a dinner for Nobel Prize winners, 1962