Yes but if the barrel has had enough abuse to cause it to split, then the rest of the action and jointing could be and probably are toast too. Life's too short to risk having your head torn off for the sake of a few quid, chop it in half and buy a new one.
No. In the UK at least you don't cut shotgun barrels in half because if you do that you're producing, or at least it can be argued you've produced, a shotgun with a barrel less than the legal minimum to class as a shotgun and that will land you in serious trouble. If you've got a shotgun which can't be sold and can't be used, you can arrange to surrender it to the police for disposal.
Otherwise, I'd agree that this thing isn't worth messing with. There are people who sleeve barrels but they are starting with a sound breech and action on a gun where the barrels are pitted to hell.
As for the rest of the thread about abusing barrels, proofing is a well worked out system conducted by the proof houses in Birmingham and London. They have a visual test before and after the proof firing, they using a specific over charge of shot and powder and they know what they are doing. I'm sure the details are on the WWW. I wouldn't assume that any old overload is equivalent to a proof firing.