"бронировать" is used more in hotel/transport business (object is place/seat). "резервировать" is used more in shopping business (object is item).
However they mean exactly the same and there is no mistake in interchanging.
"Заказать" is "to order", that is first step in purchase with delayed delivery/receipt. So, it's suitable term for reservation of room in hotel, however meaning is not the same.

"are the words "номер" and "комната" also interchangeable?" - yes.