delete question

I accept an invite and then i see the "delete" word. However, i made a move and the player replied with his move..... Is there a bug in the delete function or am I missing something???

You are given the option to delete the game until both of you have made three (?) moves.

Ok... thanks......
Now, how long do games stay listed on my displayed list? That is, I would to remove those games which are done, but I do not care to archive.

You cannot delete old games, only archive them. That's because every game (other than set piece games) are used to calculate people's ratings...your mistakes and victories are there forever!