(Q1) I will have to leave Q 1 to others.
(Q2) As far as I know, Elvis did not record "Twist and Shout". There are, however recordings readily available of the Beatles songs "Something" (see Aloha From Hawaii" CD plus others), "Hey Jude" (see Elvis Now - CD), "Yesterday" (see On Stage CD) and "Lady Madona" (see Esssential Seventies Masters Box Set).
Elvis reportedly jammed with the Beatles at his Bel Air home on 27 Aug 1965. Sadly, no recordings (if any even exist) have been released of this historic event.
(Q3) I don't know of any pictures with the Beatles, but I have attached a copy of a picture of Elvis and Jerry Lee Lewis (also present are Johnny Cash and Carl Perkins). The picture incidentally is the cover of a CD featuring Elvis, Jerry and Carl jamming at Sun records back in the fifties (titled the Complete Million Dollar Sessions).
Hope this info helps.