Do the Watchtower men have their own bible they bring in to court
It's rare these days (at least on the East Coast) for witnesses to be asked to swear on a bible. And there is definitely not going to be any swearing on bibles at a deposition in a lawyer's office. In fact, you can't be made to "swear" at all if you don't want to. As long as a witness "affirms" to tell the truth under penalty of perjury, that's enough to qualify a witness to testify.
As far as objections, what Oubilette said is right. It's not uncommon in any litigation for the majority of questions at a deposition to be objected to. As long as the attorney defending the deposition doesn't instruct the witness not to answer, it's fine. There is no judge there to referee, so they are generally just preserving objections to form. It doesn't mean that the attorney is worried that it's a damning question.