It's pretty erratic. You can see general trends by day and by week as graphs in the info tab, and there's a weekly writing exercise; you can find exactly when that is in your time zone by scrolling down a little here.
For the exercise, someone provides a prompt and everyone writes into it for ten minutes, then shares what we've got.
Well, the Stack Exchange itself doesn't support private messaging, but if you find someone you'd like to talk with off-site you can always ask to share contact info with them.
There's no formal mechanic for contact exchange; if you put yours in your profile, it'd make sense to ask the other person to use it to get in touch with you.
It can be; the system is "gamified" to reward desired actions. Depends a lot on whether you find a particular Stack that's focused on something you like thinking and writing about, and how that Stack's community behaves.
There's not so much difference, really. The "rules" in free-form games are shared assumptions rather than codified mechanics, but it's all different flavours of collaborative improvisational storytelling.