you don't have to. Men should for their faces. For women you can use soap of even hair conditioner!
No you can shave without shaving cream, but I don not recomend it. Shaving cream lubricates the skin so that the razor balde glides over it cutting the hair only. Without shaving cream you would...
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Yes. It helps greatly to reduce irritation.
No. It would just clog up your razor.