When visiting a mosque or church in a Muslim country, be respectful as a woman of what they want you to wear. You may have to only cover your hair and dress modestly, but in some cases you will need to rent a chador. They rent outside of the location for a very small fee. Both genders will be required to take off their shoes. If you don't like the custom, then don't visit the country.
I would think the Doctor brought them along to find out how they knew about him. If it was me I wouldn't give it away and just say "spoilers" and then right at the very end I would tell him that the worlds most dangerous and unanswered question is the title of a tv show.