Mozart and

Absolutely. Although Mozart knew the tune in the French version. The tune has had several lyric incarnations, including set to the English alphabet as a mnemonic device. The piece is usually anthologized on piano compilations, so you should have no problem finding it. Mozart, btw did not write the tune itself, which is traditional; merely variations on the tune.

No, you'll need to look for his variations on 'Ah, vous dirai-je maman.'

Also check out Dohnanyi's Variations on a Nursery Tune and Haydn's Surprise Symphony.

Well, let's make a distinction here between the original folk material and copyrighted versions of it. Jane Taylor apparently copyrighted her English version (now the most famous one in English-speaking countries) in 1806. But the tune itself is of French provenance; and Mozart's set of variations is attached to the French name of the tune, 'Ah, vous dirai-je Maman.' Really, think of Frere Jacques and it's the same thing. FJ has had numerous English versions and, of course, most famously, Mahler used a mordant orchestration of this tune in his first symphony. 'God Save the King/Queen' is another such song; as is 'Hail, Hail the Gang's All Here,' which originates in a Gilbert and Sullivan operetta (as I recall, Pirates of Penzance). As for the Beach Guys (or is it Boys?), well, they ripped off living people, such as Chuck Berry and Duane Eddy, so why not dead people like Jane Taylor?

Which is the same melody, and sometimes called, in English, 'Mozart's variations on Twinkle Twinkle Little Star'.

Which also features the same melody a TTLS. (A great piece, BTW).

Which is not the same melody.

There is also a wonderful set of variations for flute, soprano, and piano by Adolphe Adam. The best recording I have ever heard of it was done many years ago by Beverly Sills, Paula Robison, and a pianist whose name I have regrettably forgotten.