Mac has insomnia. Keeps waking up from sleep.

Question

I have had problems with my various Macs awakening randomly. I had a core2 duo iMac and now I have an i7 iMac, and it seems that I have a pattern. I sleep the computer and it comes on suddenly. Sometimes it goes off, but other times it stays on. I have set up the computer to go to sleep after 25 minutes: both the display and the computer. I have checked "put hard disks to sleep when possible".
I have tons of RAM and am running Mavericks. I run Parallels desktop too. I have a first generation time capsule and a 2Wire DSL modem.  The 2Wire has wireless capabilities, but I just have it feeding the time capsule with a single ethernet cable. My wife's macbook Air wirelessly and automatically backs up to the time capsule as well. I have a brother wireless laser printer and an HP wireless inkjet printer. I plug in a Denon audio receiver via ethernet to the time capsule. There is a ROKU box that connects wirelessly to my network. I don't have file sharing turned on, but I do have dvd/cd sharing checked. I don't have any other remote management protocols set up.
After I upgraded my computer to a new (used) iMac, I was dismayed to find the problems persisted. I have been scratching my head and wondering whether it could be the time capsule or the 2Wire that is the problem? I have not tried disabling time machine for either computer to see if that was the issue. I have not unplugged the time capsule and just used the 2Wire modem to trouble shoot.
I know there are certain things that can be queried via Terminal, but I don't know enough to choose what to type in. I sometimes want to tear my hair out when the computer suddenly coms on in the middle of the night, or when I can tell it has been on for hours in the morning, by touching the iMac and feeling it to be very warm.
Jim

Answer 1

Test after each of the following steps that you haven’t already tried:
Step 1
Take all the steps suggested in this support article. That's the starting point for any further effort to solve the problem. Skipping any of those steps may mean that the problem won't be solved. Note that, as stated in the article, the computer  will not sleep if Internet Sharing is enabled.
Step 2
From the menu bar, select
 ▹ System Preferences ▹ Accessibility ▹ Speakable Items: Off
Step 3
Select
 ▹ System Preferences ▹ Bluetooth ▹ Advanced...
and uncheck both boxes marked
Open Bluetooth Setup Assistant at startup if...
Step 4
Reset the SMC.
Step 5
Disconnect all wired peripherals except those needed for the test, and remove all aftermarket expansion cards, if applicable. Start up in safe mode and log in to the account with the problem. You must hold down the shift key twice: once when you boot, and again when you log in.
Note: If FileVault is enabled, or if a firmware password is set, or if the boot volume is a software RAID, you can’t do this. Ask for further instructions.
The login screen appears even if you usually log in automatically. You must know your login password in order to log in. If you’ve forgotten the password, you will need to reset it before you begin.   
Safe mode is much slower to boot and run than normal. Don’t launch any applications at first. If sleep still doesn’t work properly, back up all data and reinstall the OS. After that, if you still have the issue, make a “Genius” appointment at an Apple Store to have the machine tested.
If sleep now works as expected, go on to the next step.
Step 6
Still in safe mode, launch the usual set of applications that are running when you have the problem, including your login items, one at a time, testing after each one. Some applications may not work; skip them. You might be able to identify the cause of the problem this way.
Step 7
If sleep is still working after you’ve launched all the usual applications, reboot as usual (not in safe mode) and test again. If sleep still works, you’re done, at least for the moment.
If you still have the sleep issue after booting out of safe mode, post again.