The Phrase Finder has "Don't try to teach your Grandma to suck eggs" is older than you might think, but without any explanation of the egg sucking part.
Meaning
Don't offer advice to someone who has more experience than oneself.
Origin
These days this proverbial saying has little impact as few people have any direct experience of sucking eggs - grandmothers included. It is quite an old phrase and is included in John Stevens' translation of Quevedo's Comical Works, 1707:
"You would have me teach my Grandame to suck Eggs."
A little more on the egg-sucking part from Wordwizard:
Perhaps its meaning is getting lost in time as few people nowadays literally suck eggs. Many years ago people would suck out the egg contents by piercing the egg at both ends and then sucking on one of the ends. You could reverse the procedure and blow out the contents also. It was such a commonplace procedure then that to "teach your grandmother to suck eggs" was like a child trying to teach as new something the grandmother well knew how to do. The saying still survives despite the fine art dying out in our "civilized" and salmonella fearing culture.
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.
Best Answer
Even though marbles is a a children's game, 'losing one's marbles' was not related to the game nor the word 'marble' but a corruption of the word furniture (meubles).
Look up marbles at Dictionary.com
Marbles meaning "mental faculties, common sense" is from 1927, American English slang, perhaps [OED] from earlier slang marbles "furniture, personal effects, 'the goods'" (1864, Hotten), a corrupt translation of French meubles (plural) "furniture" (see furniture).