The symbol Cooper's Doppelganger showed the girl in hotel room
Yeah, pretty obviously the symbol in the owl cave and in the ring. Coop says he's looking for that, and we know he's looking for coordinates/locations of the gates to the Lodge. The link seems pretty obvious to me. By the way the fact he uses a card looks like another reference to Windom Earle (take a look at my post in the thread "understanding")
Yes, the Owl as depicted on the green ring - and as we've now seen in subsequent episodes, most recently, ep. 11 when Sheriff Frank Truman asks about that symbol on the old map, and Hawk says, "Frank, you don't ever want to know about that."
Is it too far a stretch to think that Audrey's white mask from the original series is the opposite version? Here's the web page banner image at a Twin Peaks blog.