On the question of period appropriateness, was Shakespeare indifferent, or ignorant? Was Cleopatra dressed in kirtle and wimple because it didn't matter or because he didn't know any better? Perhaps the answer was written on the sands of the seacoast of Bohemia, washed away by the tides, never to be known.
I'm really not so sure that Shakespeare always benefits from its loose attachment to place and time. After all, since real people are always specific to their place and time, ignoring that tends to falsify them. For instance, popular as Antony and Cleopatra seems to be, Cleopatra really seems to be some weird amalgam of King James' resentments of women (such as his mother Mary Queen of Scots or his predecessor Elizabeth I) and vamp that never carried any conviction to my eyes and ears.
For another example, consider Othello. In Shakespeare's time, the discovery of the world was in full swing. You know, Hakluyt's Voyages, etc. In that context, Othello's pompous stories from Latin authors about strange marvels, listened to open-eyed by the gullible Rosalind put a very different complexion on Othello's doubts about her. Abstracting from the setting falsifies the characters.
Nor do all aspects of Shakespeare really survive translation. Richard III really doesn't translate into thirties Fascism. It's about glorifying the Tudors by blackening their predecessor. The vision at Banquo's feast is meaningless to us. A woman's statue coming to life (I've forgotten if that's Cymbeline or A Winter's Tale) had resonances that simply do not apply today. I always found it remarkable that people so blithely ignored the basic premise of Lear that it is a cosmic tragedy if the King doesn't rule.