Hi, Vagabond,
Your title says the rifle is a Winchester Model 1905. That is a semi-auto (self-loading rifle, of which 29,113 were made by Winchester from 1905 to 1920. The caliber is .35 Winchester Self Loading, which has roughly the power of the .357 Magnum. Ammunition is no longer made.
The rod under the barrel operates the breech block for loading or clearing the gun. They were also made in .32 Winchester Self Loading, the cartridge on which the .30 Carbine was based.
The standard magazine holds 5 rounds, but ten round mags were available.
These were a little underpowered for deer size animals, but they, and the successor Model 1907, were favorites with law enforcement agencies.
Value in working order would be $500 or so in perfect condition, down to maybe $100-125 in poor shape.
Gun Parts Corp. (914-679-2417) lists some parts but not stocks. A stock can be made but might easily cost more than the gun would be worth.
Jim