Do you...have any hints or anything? I feel like I usually know who did it but have no idea how to tell what weapon it was. I'd like to get at least one right :P
EDIT: Forgot one other thing, I also don't seem to be able to actually restart with ESC
Find the body and examine it. This will give you a hint. Then look around for possible weapons... if you find one that matches the hint and is in a suspicious place, then that's probably the murder weapon.
As for the escape key issue, what platform are you on? I assume the escape key works fine during the game and you're able to open the inventory?