Does this mean my cat is comfortable around my dog?

A couple weeks ago I adopted a very sweet, incredibly calm cat who I hoped would be able to tolerate my dog, who's 60 lbs of pure playfulness. The first day or two of our trial weekend was rough - they could sit still in the same room alright, but the second the cat moved, the dog would sprint over and try to sniff/lick him (which of course, scared the cat, and led to a chase.) The cat spent a lot of time under the couch, or behind the washing machine.

They've gotten SO much better in the last week. The dog still chases every once in a while, but he's learning to approach the cat slowly, and the cat will even sit there and let him lick his face. Sometimes he rolls over on his back and sort of bats at the dog with his paws. Does this mean he's playing, or feeling threatened? I know rolling over is submissive behavior in a dog, but I've never had a cat. If he's stressed out by the dog's overenthusiastic sniffing, how would I tell? The two of them often lie around napping near each other. Does this mean the cat's stopped considering the dog a threat?