Karyn
I'd say it was totally mimed, there is accurate lip sync when he talks at the end, but the rest of the time it is clearly out, and the band is a whole beat out most of the time.  The noise you hear is just studio noise, not the guitar amps.
 
Compare to this one below.  You hear the thump as he grabs hold of the mic, and the bass player in the background is in perfect time...

Right on Karyn.  Have you every seen/heard a clip where he is live but with no vocal mic?

quantumeffect
I don't think the audio and video in the top video from the television show are sync'ed correctly.  I think I hear the clapping on 2 and 4 but see it on 1 and 3.  Also, there are two short snare rolls fairly early on in the song (at around 0:25), the first one is visually almost in time but the second one is off. 

I found a video clip of the same event that is in sync and just updated my first post with it.

At short notice I can't answer that one.  I wasn't born when these clips were recorded and I never liked Elvis once I got into music...   so shoot me.
 
All I know are the techniques used in film/tv studios in those days mainly involved ribbon mikes mounted high at 45 degrees to make use of the fig 8 pattern for external noise rejection, but they certainly weren't sensitive enough to pick out a single voice standing infront of a drum kit and guitar amp.

ok,  I take back what I said about miming,  but when he goes into Hound Dog his voice is lost almost entirely...
 
 
Incidentally, I spotted a mic under the piano.
 

This photo looks like it might be from the Elvis stage at the Ed Sullivan Theater for that show.  Elvis has the same jacket on and they are CBS curtains.  The stage is up much higher than the audience and the mic looks like it is about or above face level for Elvis.
 

Great pic Dave.  Where did you find that one?  Maybe they had a few of these hanging down but the mix sounds almost like one mic.  

The common element during that era was that recordings were just that - recordings, meaning they documented a live event.
 
You got the sound you wanted by arranging the band and the microphones and then you hit Record. There may have been some live mixing, such as bringing up a lead instrument for a solo, but for the most part it was all about getting a good sound/balance first and then trying to document it as accurately as the technology would allow.
 
Today, we have a tendency to hurry through the initial stages. Bands no longer play tunes live for months or years before taking them into the studio. Most sessions I've been in have been 50% rehearsal and 50% recording. You're just happy to get a take without too many flubs, knowing you can fix pretty much anything later.
 
That, I think, is why many older recordings have an energy that's hard to achieve with our modern, over-produced, microscopically-edited fakery.

I agree bit about the lack of rehearsal these days.  I just watch the documentary on "The Basement Tapes" and was impressed at how hard those great artists worked in a short amount of time to write and put together those songs.
http://www.rollingstone.com/music/news/stream-the-new-basement-tapes-featuring-elvis-costello-and-jim-james-20141105
 
Not all of us are that talented but I was impressed at the energy they put into it all.  Relating back to Elvis or groups like his, when you consider the song, lyrics, singing, playing, and arrangment for playing their songs live back in that day, there really was a huge amount of talent coming from the artists compared to the audio/studio production.  That is what I like most about the old stuff.  

I just did a search that included ... Ed Sullivan, Sound Stage , Elvis ... and then carefully waded through all the search result images (I was interested myself).
 
There were a number photos from other Ed Sullivan Show performances that showed an overheard mic.  This was the only one I found that (I think) was associated with the performance that Rimshot was asking about and showed any sort of mic placement.

Here is an article that might help...recording Ricky Nelson in the 50's. http://www.prosoundweb.com/article/inside_the_techniques_used_to_record_rick_nelson/
 
Ben