How can I download something with display turned off (Ctrl+Shift+Eject)? In Lion I could just press Ctrl+Shift+Eject to turn off display on iMac. 

In Mountain Lion it looks like this shortcut turns off whole computer. It looks like sleeping mode. So my downloads stop when I press Ctrl+Shift+Eject. 

How can I download something with turned off (Ctrl+Shift+Eject) display?
I always use this combo as well, however this does not immediately sleep my machine in ML.

I believe this is a separate issue at play here. According to Siracusa's ML review, unlike in previous OS X versions, in 10.8 your system is eligible for sleep immediately upon "user inactivity" (the inactivity delay being set in your energy saver settings) unless the currently running apps make "power assertions".

In this case, that key combo is signalling to the OS that "user inactivity" should be triggered right away. In such a case, if your particular app that's downloading stuff does not make power assertions, your system will sleep.

I have come across the same issue generally (not necessarily related to triggering the screen sleep combo). See: Mountain Lion sleeps too aggressively