Learn English – Words Inspired by Television

The New Partridge Dictionary of Slang and Unconventional English (2008) says:

have a cow to become emotionally overwrought; to lose control US, 1966.

Speaking of Animals: A Dictionary of Animal Metaphors (1995) by Robert Allen Palmatier says:

HAVE A COW to have a cow. To have an anxiety attack. Source: COW. WNNCD: O.E. On the TV show "The Simpsons," Bart Simpson says "Don't have a cow, man!" meaning "Don't get all upset about it." Bart is likening an anxiety attack to giving birth to a cow - a frightening thought. Normally cows are the ones that give birth to cows - i.e., bull calves and heifer calves. Compare Have Kittens.

WNNCD is Webster's Ninth New Collegiate Dictionary (1983) and O.E. means Old English, but the O.E. must apply to the plain word cow rather than the phrase. (The OED dates cow to Old English.)

This Yahoo Voices article - Idioms Unpacked: "Don't Have a Cow" - also claims it means to (not) give birth to a cow, which would be distressing for a human to do. It lists a number of references at the end, but I've not followed them.

A quick search of Google Books shows this snippet dated 1962 from Field and Stream, Volume 67:

"Oh, don't have a cow," Chip said confidently. "They just haven't begun to fly yet."

"If they don't fly soon," Andy insisted, "they're going to need landing lights."

(Care must be taken with Google Books' snippets as they're often mislabelled, but following the story text we find an advert for a "NEW 1963 book of homes", so it's likely from 1962 or 1963.)

Searching Subzin.com, the first film I found to use the phrase was Sixteen Candles (1984):

00:39:00 I don't know, Jake.
00:39:02 I'm getting strange signals. Well, they're not comin' from me.
00:39:05 Everything's fine. Don't have a cow.
00:39:08 Okay. 00:39:10 Just remember one thing.

Edit: Good timing, as the OED have just released an update to the dictionary containing the phrase for the first time. The first quotation is from a 1959 newspaper:

1959 Denton (Texas) Record-Chron. 26 Mar. 3/2 He won't let me watch rock 'n roll shows... He'd have a cow if he knew I watched 77 Sunset Strip.

In my opinion, the adverb enough used in this sense creates a conversational rapport between the speaker/writer and listener/reader. In strangely enough, interestingly enough etc., the enough might mean "enough to go so far as to consider it such". The reason it's used is perhaps a fear that the addressee might not feel the described circumstances really warrant the qualification strangely/interestingly/etc. By using it, the speaker/writer acknowledges that their listener/reader is right to think that, but that the circumstances really can be considered such for the sake of their conversation.

In any case, this enough is not used in a formal register. When it is used in a formal setting, it signals a (temporary) lowering of the register.

Just my two cents.