In The Springfield 1903 Rifles, Brophy says there were 2,330 M2 rifles made in FY 1933. I would say your rifle was made in late 1932 or early 1933. I shot an M2 Rifle in ROTC years ago. I know we loaded them single shot but don't recall if we pushed the cartridge down into the magazine without removing the magazine from the rifle or just by pushing them into the chamber. I don't think it will hurt the extractor to slide over the cartridge head but am not certain about that. Rather than risk breaking a hard to find part I would load the cartridges in the magazine. The locking lug in the photo appears to have the flat surface like the late M2 locking lugs. It just occurred to me that some replacement bolts for the 1903 from WWII have the square locking lug. The original bolt on your rifle would have the concave outer surface.
The M2 magazine will a very noticeable M2 stamped in it. Your magazine must be for the 1922M1. Since it does not come up quite as high in the receiver the M2 bolt may over-ride the top cartridge in the magazine. If it picks up the cartridge and loads in reliably then you have no problem. It is just not the correct magazine for the rifle. The rear sight on your rifle needs the screw in aperture. They are, or at least in recent years were, available from Lyman. They are a $10 to $15 item.