Sorry to say, I don't know any specific names for the games you've described. I'm familiar with the hiding an object in your hand behind your back game, but as far as I know, it doesn't have a name. Any time I played it as a child, the other person just said "Guess which hand?" or "Pick a hand."
If I'm understanding the second part of your question, there are many different games children use to choose a player. Probably the most common would be "eeney, meeney, miney, moe." You point to each person as you say each word in the rhyme, and the last person/word gets eliminated.