Answer to #1: False. Unless you leave the CD player on another unit that tends to get hot. The CD player should always be on a shelf by itself. You don't think that you have to put away all one hundred CD's in a hundred disk changer every time you shut it off do you?
Answer to #2: Absolutly false! The only thing that might cause damage is shutting the device off while it is playing. That could damage the unit by screwing with the drivers that move the laser.