What is this part called? (Photo on Mosin Nagant)

[Bang]

^^^^ Correct. Some had them, some didn't, and there were differences as noted. Kind of like never buy a car that was built on a Friday.

[littlefitz]

I'm pretty sure that's the shoulder thing that goes up.

[carrguy1]

As someone mentioned they*re called escutcheons but also as discussed not needed. Check this out. http://7.62x54r.net/MosinID/MosinFeatures06.htm

[sawgunner73]

I wouldn't add the grommets to the rear of the stock. These guns had all types of stock variations, and that could be a rare configuration. I would research it further before you potentially damage the collectability of the rifle.

[pacomdiver]

wartime ones didnt have them. a way of speeding up the production process, sometimes they had a half liner in the front slot. the ones that were there are from being refurbed and an older stock put back on after refurbed