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take off the forend, and remover the barrels. It should be on the action flats, the barrels, and maybe both the forearm wood and metal. My 1909 Ithaca is like that.

Never seen one with a goose (or any animal) on the stock. Probably done by a previous owner.

Early Ithaca barrels were "Damascus" (also known as stub twist), and fit for black powder only. Later (early 1900s) they went to "fluid steel" which was suitable for smokeless powder. There is an improved later version of the barrel steel but I can't remember the name right now.

Does the gun have double triggers? (most likely), does it have ejectors, or jut extractors? How many positions on the safety? Splinter or beavertail forearm?

Old Ithacas (standard grades) don't bring the buck like Parkers or some other doubles, but they are fine shotguns. If its in real fine shape, $400-500 maybe. If worn, not nearly as much.
Ejectors were an option (cost extra), but will add to the value of the gun, somewhat.

See if a dime will balance in the muzzle. If it does, that barrel is full choke.

If in sound condition, use only low brass (high base) shells. Use ONLY LEAD SHOT!!!!!!!!!!! This is important. The guns made before the 1950s when plastic shot cups became standard in ammo, are choked much tighter than newer guns. There is more constriction in the choke, and steel shot can bulge or even split the barrels of the old guns. It just doesn't compress the way lead shot does, when it goes through the choke.

If you are shooting it, I would recommend 3 or 3 1/4 dram equivalent loads. The "express" 3 3/4 dr eq. shells will be hard on the gun, and may crack the reciever, over time.

I have a 1949 letter (sent to my grandfather) reaffirming the guarantee on the springs, which also states not to use the express shells.