I just looked it up and you're technically right; that's apparently the official dictionary term, but I've never heard anyone actually say that. Everyone calls them "soshoku danshi."
A Google fight (excluding Chinese) turns up 549,000 results for 草食系男子 and 571,000 for 草食男子. Much closer than I expected, but the 'kei' is definitely not required.
"Soshoku danshi" is literally "grass-eating (herbivore) man". "Soshoku-kei danshi" is "grass-eating-group man". The latter is slightly more correct, but both are in common use.
You are correct. The calligraphist got the inclusion of the "kei" right, just not the article text. I don't think that "soushoku danshi" is wrong, though. Also, sometimes just "soushoku doubutsu" is used (literally, "herbivore").
I don't think it's literal; it just uses meat eating as a metaphor for aggressivity and drive. Just like in English we have "carnal act" for sex, referring to the flesh. Or whatever.
It's not even a word denoting vegetarians, but rather herbivores. A vegetarian is called 菜食主義者 (saishoku shugisha), not "soushoku" anything. "soushoku danshi" comes from "soushoku doubutsu", or "herbivore".
