iMac 21.5 Inch Mid 2011, OS High Sierra 10.13.6

HI nitamarie.

This statement caught my attention:

I tried turning "off" at night, but started having fan problems - fans would run forever,

That could be something else at play. The System Management Controller (SMC) deals with lots of fiddly bits including fan behavior. An SMC reset is easy to do, can't hurt anything, and might cure both the "wakes-at-night" issue and the fans. Please see:

How to reset the System Management Controller (SMC) on your Mac - Apple Support

Your iMac does NOT have the T2 Security chip, so use the instructions under this heading:

How to reset the SMC on Mac desktop computers

When it says to wait 15 seconds, it means 15 seconds. I've counted in my head too fast before and the reset did not "take." Too long wont hurt anything--too short won't perform the reset properly.

FWIW, I gave up on the Sleep function years ago. Ever since the inception of OS X there have been Sleep related issues. And even more lately. I just use Display Sleep and keep it set for 15 minutes so I'm not burning the screen. That way your Mac uses very little power over Sleep and is ready to use at the touch of a key without any grogginess. UNIX based systems were made to be always on and they’ve been running busy servers 24/7 for decades.

In Sys Prefs>Energy Saver, is Wake for Internet Access checked?

You're welcome. The problem may be "Wake for network access" as BDA indicates, you can try that first.

You're welcome.

Very likely the problem. :)