Learn English – Look who’s talking

I would describe it as wordplay through the use of puns.

While not exactly the same, the following are all examples of Tom Swifties:

• "Pass me the shellfish," said Tom crabbily.
• "That's the last time I'll stick my arm in a lion's mouth," the lion-tamer said off-handedly.
• "Can I go looking for the Grail again?" Tom requested.

A Tom Swifty (or Tom Swiftie) is a phrase in which a quoted sentence is linked by a pun to the manner in which it is attributed. Tom Swifties may be considered a type of wellerism.

The name comes from the Tom Swift series of books (1910–present), similar in many ways to the better-known Hardy Boys and Nancy Drew series, and, like them, produced by the Stratemeyer Syndicate. In this series, the young scientist hero, Tom Swift, underwent adventures involving rocket ships, ray-guns and other things he had invented.

A stylistic idiosyncrasy of at least some books in this series was that the author, "Victor Appleton," went to great trouble to avoid repetition of the unadorned word "said"; elegant variation used a different quotative verb, or modifying adverbial words or phrases. Since many adverbs end in "ly" this kind of pun was originally called a Tom Swiftly, the archetypal example being "We must hurry," said Tom Swiftly. At some point, this kind of humor was called a Tom Swifty, and that name is now more prevalent though incorrect.

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.