UPDATE: I solved the problem day after it happened. I simply set compatibility mode with Windows XP SP3 and that did it. Game started working well. All tho checkpoint saving or manual saving takes a bit longer than it should but still, it worked.
So yeah, I did not had this problem on previous Windows, like Windows 7 or XP, but on Windows 10 sometimes FEAR just crashes back to desktop while saving the progress automatically on the checkpoint. For example, I had a crash in the moment after you meet that fat guy second time, when he is hiding with holding cactus above his head. And so, after you talk to him and move down the corridor, game starting to save the progress and then it freezes and crashes without any error whatsoever. I don't know why this is happening, but again, like I said, I did not had this problem back in the days on previous Windows OS. Is there any solution to this?