Hmmm...up to you. I would eat it and probably serve it to my family but I wouldn’t put it on a cake I was selling to a customer...
I always keep my left over frosting in the refrigerator or freeze it. I am anal retentive and leave nothing at room temp.
I would just throw it in the fridge. Unless it is has been applied to a cake for personal consumption (because I don't like cold cake and when I want a slice of cake I want it right this second and don't have the patience to let it warm up 
) I wouldn't leave it at room temp. When you need to use the frosting just pull it out of the fridge and let it soften, then give it a quick re-whip and it should be good to go.
I warm my cake slices in the microwave for about 10 seconds. Perfect.
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