This is by no means the definitive answer to any part of your query - but here goes. I will break it down for ease of writing and reading:
Beard: The beard you see Elvis sporting is a direct consequence of his film career, or more specifically, one film. In 1969, Elvis appeared in Charro, a gritty western in which his character was required to have a beard. Elvis was apparently uncomfortable with the idea of this and/or the physical issues of having a beard (i.e. itchiness!) so various members of his entourage supposedly grew their own to help him out! Even Colonel Parker got in on the act! (To the best of my knowledge, Parker usually went clean-shaven, but you can see him with some facial growth in a single image used in "Elvis by the Presleys"; I guess it's confirmation of the "everybody making Elvis feel at ease" anecdote...). The only other times Elvis had any sort of growth, to the best of my knowledge, is when he came out of hospital for various reasons.
Shaving: Elvis preferred electric shavers, like my good self. Although they can give you a slight rash on the softer skin (i.e. neck) from time to time, I like the "no nonsense" aspect of them. I can't be dealing with shaving foam and the delicate precision needed to avoid giving yourself a nick! That said, Elvis could have let other people shave him if he'd wanted to. I guess he didn't trust any other SOB with a razer to his face! (A logical enough fear ). There are a few photos here and there where he appears to have very, very mild stubble; but this seems more a consequence of his skintone being lighter in those shots (i.e. providing more of a contrast for the stubble to be visible) than anything else.
Androgyny: I think it's fair to say that Elvis was not a conventional-looking man. He always wanted to stand out in the visual sense. Even when still dirt poor (well, maybe not dirt poor, but still pretty poor), he began kitting himself out in the best and brightest stuff that Beale Street and black places had to offer - two tone shoes, bright pants and shirts, loud jackets etc. A favourite combination of his in the 50s was pink and black! Elvis always stayed with the times, and as the years progressed, so did his personal fashion. In the 70s, he really pushed things by donning furs and capes and equipping himself with canes. Cosmetically, he was something of a trend-setter, too (though only more recently do you see men embracing stuff that Elvis was doing daily 50 years ago!). So yes... carefully coiffed and dyed hair, mascara and lip balm were all part of the package. As if to compensate for this non-masculine image, he was a pretty coarse guy - tough in a fight, quite profane and "one of the lads" (hence all his practical jokes and wild spending sprees where EVERYONE got a horse, or a car, or a fancy wristwatch or whatever).
Chest hair: I'm not sure on this one. From what I recall of a picture of him and June in Peter Guralnick's "Last Train to Memphis" (from, er... 1955/56?), his chest was completely smooth. Fast forward to his army induction in 1958 and I seem to recall a little growth. Even if my recollection is wrong there, he definitely had little "flecks" ten years after that in the Comeback Special. Bottom line: I don't recall ever seeing obvious amounts until well into the 70s era. Remember... it was quite fashionable to have chest hair at that point in time! Now, I don't want to fill you or anyone with bad mental images here, but speaking personally (you may look away now!)... my own, erm, fibrous filaments have continued to grow since I hit puberty. I SWEAR I've gained a little more between my 21st and 22nd birthdays! It's not as bad as Elvis' in the mid-late70's, but it's certainly more than he was showing at my age. ****. Smooth chest people: I hate you.