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I was wondering, if I want to make sure that my eggs white are safe to eat in my royal icing, can I heat the e.w to 140 or 160F then cool it completely to make my icing or will that cause a problem?

Hello sylvia57

Best way to pasteurize egg whites is to do them in the shells on very low heat for longer. Egg whites begin to cook at 140F so the slower longer method is more desirable. Temperature control is extemely important which makes this process tedious for home pasteurization.
Pasteurzing egg white: http://www.wikihow.com/Pasteurize-Eggs
If you don’t want to do them at home, pasteurized egg whites are available in supermarkets for about £3.00 for a 100g carton (UK) Use a little lemon juice or cream of tartar to stabilize the whites, most recipes will say how much to use.

So we can use carton egg whites to make royal icing. thank you

Yes you can but they do benefit with a little cream of tartar to stabilise them.