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.
The metaphors are straightforward, I think. A lily is a symbol of purity, in its whiteness and "cleanness." A virgin is a lily; a slut is coated in mire. (Mire, as in the couplet "muck and mire," is a wet admixture of clay and dirt, which if stepped into will be difficult to emerge from. If you're wearing galoshes, when you struggle to free yourself, your feet and your galoshes will likely part ways, given the sticking power of the mire.)
So emerging from the mire is a step in the direction of purity, but by no means a big step. Hence, the contrast between the lily and the mire is more ironic than realistic. In other words, the person who emerges from the mire is a very long way indeed from purity! The figures resemble a left-handed compliment. Suppose I ask a friend of mine how good-looking was his blind date last night. He responds, "Well, she wasn't a dog, but she wasn't exactly a hotty, either." That sort of thing.
Best Answer
One of the definitions of hashed in your reference is "mangled" (combined in a haphazard way; contorted and damaged thereby).
The metaphor is describing how time passes: it "tripped by" — that is, with dainty dancing steps — on "rosy wings". While time does fly according to one turn of phrase, the juxtaposition of flying and stepping is what causes the mixed metaphor.
With regard to haste in writing, it has nothing to do with the metaphor itself, but with including it at all. Because of time pressure, Porter had less opportunity to self-edit and eliminate the mixed metaphor.