I would like to know if I make a rosette cake out of Italian meringue buttercream, if I can refrigerate it?  Will the rosettes harden like it does when you put a container of leftovers in the fridge?

Yes, the buttercream will harden up but them in order to serve it you just set it out and let it come to room temperature...maybe 30 minutes to an hour depending on the size of the cake and the temperature in the room.  You want it to come to room temperature gradually.