I have three characters in my scene that enter and leave at different times. How do I set a key frame so a new character does not enter the scene until midway? Even though I don't pull it in until the middle of the scene it still pops up at the beginning.

The best option is to make the character invisible until needed.
Check #3:
http://manual.reallusion.com/iClone_7/ENU/Pro/Default.htm#iClone_7/Pro_7.0/09_MotionPlus/Visibility_in_Perform.htm?Highlight=visibility