There is no keyboard shortcut, but you can initiate it using this command line:
mc15.exe /MCC 23000
Auto-Import works fine with sleep mode enabled on my systems. However, I rarely-if-ever let my machine sleep with MC still open, so I can't say for sure that I'm not seeing the same behavior. My Library Server system now runs the auto-import itself, and it is set to never sleep (because it is always doing something, it has many tasks).
I used it directly on my HTPC for years though, and it is set to go into full Hybrid Sleep. I never noticed any issues. Again, though, I generally close MC on that system when it isn't in use, so I may not have encountered this specific issue.