when to use adjective as adjective or noun as adjective? For example, it can be stated as "cancer center," but what about "cancerous center?"

I know most people would use " cancer center," but should the "cancerous center" also be right?
No. All people would use cancer center. Cancerous center is something different, like a building which is infected with cancer (if that were possible).

Here's another example. A beauty shop is a shop which provides and deals with beauty. A beautiful shop is beautiful.

The question is whether you are describing the noun or defining the noun's area of responsibility.
I'd say it depends on what you mean. I would interpret "cancer center" mean a building or group of buildings where cancer is studied and/or treated. If I wanted to refer to the center of a cancerous mass, I would say "the cancer's center" or "the center of the cancer." "The cancerous center" would mean, to me, the cancerous center of something which was not entirely cancerous. (For instance, "Some of the healthy tissue had to be removed in order to be sure of removing the cancerous center.") They are quite different in meaning.