Back in December I switched from Eclipse to IntelliJ IDEA but within a couple of days I switched back. Today I tried IntelliJ again and basically I have only one issue left.
I ended up creating a Macro which does the Save and the Make Build. I then assigned the Ctrl+S shortcut to this Macro and then it works fine.
What if you just remap the keys from Ctrl+F9 to Ctrl+S? Basically you will just keep up with the habit of pressing Ctrl+S since you do that in Eclipse - just a matter of pressing a different key in IntelliJ.
IntelliJ IDEA 12 indeed comes with out-of-process compiler which compiles the project classes automatically, but it seems to be unpredictable - you might refresh the page and the class is not yet compiled. So in the scenario with JRebel the off-process compilation is no use, unfortunately.
This may not be exactly what you want but normally I leave auto-save on, and when I need to compile the current class I'm working on, I just do Ctrl + Shift + F9.
My IntelliJ IDEA version is 2017.3.4, and I modified some values so that it compiles automatically.