In addition to Viola, Imogen in Cymbeline becomes page to the Roman general Lucius and accompanies him into battle, so it would certainly be plausible to have her handle a sword in performance, even though it’s not specifically mentioned in the text.
Cymbeline also has a totally goofy ghost scene in which all of Posthumus’s deceased family members pay him a visit, so it provides yet another answer to #3.
Any scene in Antony and Cleopatra that isn’t set in Rome (and Cleopatra is plenty seductive).
Do they have to interact? If not, Mamillius and Perdita in A Winter’s Tale would qualify (but he dies shortly after her birth, so they don’t have any scenes together). And there are Anne and William Page in The Merry Wives of Windsor, but they don’t have much onstage interaction either. Imogen has a rather sweet scene with her two long-lost brothers (seriously, Cymbeline has everything!) although they don’t know they’re siblings at the time. Of course, none of this is exactly “well-known play” territory…
Do you want an interpretation that is so cliched and just-plain-dumb that most actors would shudder at it even though it’s popular with the general public, or one where there are multiple, reasonable ways to play it, but people tend to be so convinced theirs is the One True Way that they’ll fight over it? I’d vote for Oedipal Hamlet for the first one, and “exactly how sexist (or not) is The Taming of the Shrew?” for the second.