I seem to have stumbled across a solution that works for me. I noticed today that when my wife was logged in, she was able to wake up the iMac from sleep with just the mouse our keyboard. This led me to test some of the programs I have running. It turns out that Mesh (file sync, remote access) was causing the problem. Now that this program is not running (I had it set before to run at login), the sleep and wakeup functions are working fine.
One note, as stated in my previous post, the Mac store replaced my logic board when trying to fix this problem. I cannot be sure if the new logic board was part of this remedy or not.
The logic board did make some difference that I did notice before I figured out the problem with Mesh. Before the repair, when playing music with iTunes and the display went to sleep, in order get the display to come back on I had to put the iMac to sleep by pressing the power button (therefore causing the music to stop). Then I woke up the iMac and had to restart the music. Now, when playing music and display sleep kicks in, moving the mouse or keyboard press will wake up the iMac (and not interrupt the music).