The second half sounds familiar to me.
This thread reminded me of the clapping game. Do kids still play that? I remember only the start of one of the songs, See See my playmate, come out and play with me. . .
We had a bunch of songs that went along with chinese jump rope but for the life of me I can't remember any of them.
And there was that string game too, where you'd make patterns by wrapping string around your fingers and hands then you pass it on to someone who'd make something else and so on.