We say 'hide and seek', because first the person hides and then you try and find them :)
Hotchscotch? In the UK we call it 'hopscotch'. I thought it was the universal name for it but perhaps not.
You're talking about a skipping rope right? Not quite the same as an elastic band. Anyway I guess 'skipping' is the game you mean. In the US they call it a jump rope instead.
In the UK a popular game is called 'tag'. The idea being a person is 'it', and they then have to try and run and catch other people. If they touch a person and say 'tag' then that person becomes 'it'. The game continues like that, with the person who is 'it' being swapped when they get caught.
Another one is 'British bulldog' or 'bulldog', when people line up on one side of the playground, and then try to get to the other side. Again a person is 'it', and they try to stop people getting to the other side of the playground. If you are caught you then are out, and have to join the people trying to catch the others. Eventually you end up with only one person left who hasn't been caught and they are the winner. I used to play this game at school, but it's quite dangerous so some schools ban it.
And of course in the UK boys always like to play football in their free time on the playground.