Yes, they do. If you are considering 'bread and butter' as a single item, then it's fine to use the singular form. It might sound more natural to say either 'Bread and butter is my favourite breakfast' or 'Bread and butter is my favourite thing for breakfast.' If you are considering 'bread' as one product and 'butter' as another, then you use the plural form.

Well, no, they are not used the way you used them. One's bread and butter (it has the position of a noun) refers to a person's job or something that he/she does and earns money for a living: E.x: At the moment teaching is my bread and butter. Or it could refer to something ordinary: It's a bread and butter exam so sweat no sweat! You could also check this link: http://dictionary.cambridge.org/dictionary/british/bread-and-butter