As an ump who does occasionally forget the count ... I would never ask the book for anything other than the score — the only thing I don’t typically keep.
The home book is the “official book” for stats — that does not make them an official. They are team personnel, and borderline at that. Same goes for the scoreboard operator.
When you pay one ump, you get one ump.
Interesting. I have seen umps consult with scorekeepers on counts and outs more than a few times. That said, it's most often in games that count for little to nothing (showcase or friendlies).