No. The high sugar content will help prevent spoilage. Keep unused icing in the fridge for up to three weeks, then toss. The icing consistency will change and be difficult to work with.
I agree. I've had it sit for up to 2-1/2 weeks and it was still good. If you are really concerned, Toba also recommends (in her book) using water instead of the milk. Although i've not tried it, it may very well keep in the fridge longer.
The only time I used water, I had trouble with colors bleeding together. It could've had nothing to do with the water, and everything to do with the colors I used, but I went back to using skim milk!