The Beatles idolized Elvis--particularly John and Paul. Elvis wasn't a writer, but he was a consummate performer and vocalist. His greatest stuff is from the Sun Sessions--his Mystery Train and Blue Moon of Kentucky are incredible. Some of the early RCA is great too, but by the 60s he was toast. He couldn't reinvent himself.
In some ways, the Beatles were the perfect response to Elvis. To his "hot" style and pelvis gyrations they played it cool. The Beatles did their cowpoke kneebends and that was about it. Though they could do rock shouters like Money or Long Tall Sally, there was always a little bit of irony, ice, to it.
As far as I know none of the Beatles ever covered an Elvis song. Even on John's Rock and Roll album. Maybe a bridge too far . . . ?
As far as the Elvis name, it's Norse. All-wise. El-vis. I've always doubted that Vernon and his wife had Elvis explicitly in mind when they named Elvis and were probably thinking of the more popular Elvin and just sorted screwed up. Or thought they could make it "better"--who knows. I don't think Vernon Presley was a Norse language scholar or anything. Elvis Aron Presley is his full real name. They couldn't spell Aaron right either.