Can I stir my ice cream mixture over an ice bath to cool rather than wrapping and refrigerating or does it need to 'rest'? Thanks!

You can certainly use an ice bath for cooling the custard. Make sure you get it to 40 degrees or lower before spinning it. That said, your final ice cream will have a more luxurious, supple texture if you can refrigerate the custard overnight. The protein strands will relax and absorb larger quantities of liquid. The resultant consistency is sort of amazing.

The ice bath will do fine for any sort of mixture I know of.