Do older dogs play less than younger dogs?
While this is true of wild animals who have to work harder to survive and to thrive, older dogs play a lot when they can and we really need more data on this question.
Do dogs have a theory of mind?
We don't know. While some studies suggest they don't, we need more "naturalistic" research especially when dogs are socially interacting. Because play is a foundation of fairness and there is a good deal of cooperation among the players as they negotiate the ongoing interaction so that it remains playful. Perhaps dogs even know what their playmates are thinking and feeling. Do they have a theory of mind? While I think so, we still need more data on this aspect of play as well.
Why do dogs roll and writhe on their back?
It could be to impart an odor. A wild canid known as the raccoon dog who lives in South America has a scent gland on its back. Dogs might also roll on their back to mask their own odor. And, of course, it might feel really good so why not do it? I love watching dogs writhe on their back and they look like they're in doggie heaven.
Do dogs have a sense of time? The "two minute warning."
We really don't know much at all about the dog's sense of time. Yet, people often use what I call the "two minute warning" and ask their dog if it's okay if they leave in 2 minutes, or people tell their dog something like, "You have 5 minutes more to play with your friends before we go to the store." They also ask their dog, "What the hell took you so long, I've been calling you for minutes?" or "Where were you when I called you?" I can well imagine the dog thinking something like, "Huh?"