If not marked, you will need to have it examined by a competent gunsmith
Winchester bought Cooey in 1961.
cooey 410 gauge model 840 Winchester Canada. single shot
150 - 200 bucks in .410
Typically, no. There are some .410 shotguns which can also fire one or more rimmed pistol cartridges (usually .45 Long Colt, but I've seen them for .41 Magnum and .44 Magnum, as well), but these are purpose built to be able to do this - it cannot be done in an ordinary .410 shotgun.
No, you cannot shoot .44 Magnum and .410 shotgun shells from a Thompson Super 14 interchangeably. The Thompson Super 14 is designed specifically for .44 Magnum cartridges, while .410 shells require a different type of chamber and barrel configuration. Attempting to fire the wrong ammunition can result in dangerous malfunctions or damage to the firearm. Always consult the manufacturer's specifications for compatible ammunition.
Numrich gun parts
No they did not. Cooey only made single shot shotguns starting in 1948. For awhile, back in the late 30's, they assembled shotguns for Iver Johnson in Canada.
No. But there are revolvers made to shoot .410 shells that also fire .45 Colt cartridges,
Not safely. Despite the tales you have heard about this, the pressures of a .44 Magnum are well above the design pressure of a .410. WHEN (not if) the gun fails, someone is going to be hurt. No. Don't.
3 inch
No, you cannot use a 3-inch shell in a Cooey Model 840 .410 gauge shotgun, as this model is designed specifically for 2.5-inch shells. Attempting to use a longer shell can result in safety hazards, including potential damage to the firearm and injury to the shooter. Always consult the owner's manual for specific ammunition recommendations.
Probably. Check the side of the barrel. If it is marked "2 1/2 - 3 inch" you should be good to go.