So I cannot holster weapons properly for some reason. The weapon sticks on my player's body when I try to holster it. Agent 47 continues his animations as if he has holstered it. I cannot put away my sniper rifle (or any other rifle) in missions in which stealth is important. I will put up a screen in a second to show it.

I'm pretty sure you can't holster larger weapons like sniper rifles. Are you referring to the petersburg mission? Because I can distinctively remember having to pick it up, drop it, pick it up, drop it over and over again to get it through without any of the civilians immediately freaking out... Contracts & Blood Money are still great, but I don't think Silent Assassin has aged very well at all. You're not even allowed to run without bringing about suspicion.

I had a guess that was the case but holstering large weapons looks so weird.

Yeah, he's right, you can't magically put a rifle in your pocket. I recently started playing it as well and damn, some of those early missions are tough if you wanna get through with a good rating. I'm stuck in Kirov Park.