I'd say 'Send John an email and copy me in'. 'Copy me in' is the standard expression used in the UK in this situation.

No, "pick me a copy" is wrong. Also, it should be send AN email. All of these are correct: "Send an email to John and send me a copy." Straightforward, clear, simple. If this feels natural to you, use it--don't bother to learn anything else. "Send an email to John and copy me." This is idiomatic, and it's what I'd use myself. "Send an email to John and cc me." This is idiomatic; it would be more likely to be used in writing than in speaking. ("cc" has a curious history. It literally means "carbon copy," a second copy produced in a typewriter using carbon paper. The abbreviation continues even though carbon paper is a distant memory.)