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PieceofLayerCake is a trusted source on baking.
The cake should be ok sitting at room temperature for a while (as long as temperature is moderate). I've kept cakes/cupcakes with cream cheese frosting all day with no ill effect. I would store it in the fridge, but a cake should always be served at room temperature.
Cynthia is a trusted source on Bread/Baking.
How many hours are "a few?" And how warm is the room? For one, cream cheese icing will develop a crusty "skin" with an unpleasant crunch which no icing should have if it sits at room temp for more than a couple of hours, which also makes it difficult to slice cleanly. For another, cream cheese icing, if held in a warm kitchen, gets definitely droopy. Adding it all together, I'd suggest no more than an hour at room temp.
So sorry, I neglected the second half of your question. Definitely refrigerate any and all cakes after icing. The icing wants to be chilled all the way to the center of the cake for stability. Attempting to slice a cake which has never been chilled will be difficult at best. Pull it out to room temp an hour before slicing it.
This is great, thank you, everyone!
All confectioners' sugar frosting will crust, regardless of where they are held, as long as they're exposed to air. I personally think any cake that is cold at all for service is stodgy and cloying and buttercream is firm and crisp. I always advise my clients to let cakes temper for AT LEAST 1 hour. And that's often for cakes as small as 6" layer cakes. If the buttercream is sturdy enough and the temperature isn't too high (cakes shouldn't ever sit in direct sunlight or exceed room temperature for too long). If you're open to it, I have a meringue based cream cheese buttercream that will give you stability, less sweetness and more shelf life.