I'm assuming these are two separate phrases. Yes, they both make sense. "Any hurry?" does not mean "is anybody in a hurry?" "Any hurry" means "is there any need to do this quickly?" "Any need to rush?" "Any need to expedite this?" For example: "When will it be ready?" "Any hurry? Usually it takes 48 hours. We can do it faster but we charge extra for rush orders." "It's customary for me to start off by saying..." sounds like the start of an informal speech. It means "I usually begin by saying..." For example: "It's customary for me to start off by saying 'welcome' to our visitors, but today we are privileged to have friends from South America with us, so let me begin with a warm 'Bienvenido...'"