Is it OK for an adult to join or start a game with some unfamiliar children (if they don't object)?
This all depends on where you live. If you live in some countries, where they adore their kids, then this would be considered par for the course. I remember walking down a street with our (then) three year old daughter in one such country. A waitress came out of a cafe, picked her up, and carried her into the shop. 30 seconds later they re-emerged, our daughter sporting a freshly made ice cream.
The same thing just wouldn't happen in the UK.
The rule I would use is: if an adult is present with the kids, and you have made enough eye contact with the adult to know that they don't mind, then by all means join in or even start a game. If however the child is on their own, then I wouldn't initiate a game. If they initiated one, by kicking a ball to you, or whatever, then by all means kick it back, but be wary.
There are two real issues here. If you live in a country / neighbourhood, where it may be dangerous for kids to play with strangers etc, then by initiating any such contact, even with the best of intentions, you are encouraging that kid to play with strangers.
The second issue, is that your actions may very easily be misconstrued. The other day, I stood in our daughter's playground with all the other parents, waiting for the bell to ring. A little girl (I am male) had found a tennis ball, and decided that I would be a suitable goalkeeper. I had no idea who her parent was, but kicked the ball she rolled to me back to her. This continued for a few minutes until her mum noticed she was playing with a stranger, albeit, a fairly reluctant one. The mum told her daughter to play somewhere else (I noticed with some sad irony that she didn't actually volunteer to play with her herself, but continued yakking on her phone). At that point I felt fairly uncomfortable, despite the fact that I had stood in the same playground for the last four years.
I'm not saying the second point is how things should be, I am saying it is how they actually are, at least in my neck of the woods. Its a sorry state of affairs, and I always love going abroad to places where kids are treated with so much more warmth and friendship than they are in the UK.