I was wondering about something (where Harry got his broomstick after getting the diadem, since his Firebolt was gone), so I looked it up, and I looked up the Bathilda/Nagini scene as well.
Turns out Bathilda/Nagini said "Come"
before Harry went upstairs, not after: (p. 336)
"Come!" called Bathilda from the next room.
Hermione jumped and clutched Harry's arm.
"It's okay," said Harry...
So my question is still:
3. How could Nagini (in Mathilda's body) say "Come" from the next room?
Either Nagini somehow manipulated Mathilda's vocal cords (unlikely) or she spoke Parseltongue.
So
why didn't Hermione recognize that it was Parseltongue? She's heard Harry speak it on more than one occasion. Surely she would do more than just jump, as smart as she is.
Obviously there's no way around it, since it would ruin the reveal. JKR could have just left it out though, and had them follow her (since they wanted to talk to her anyway).