The word 'any' has different uses, and the word that follows it depends on how 'any' is being used. In your first set of questions, only the first - 'Have you got any questions?' - is correct. This is because 'any' is being used as the plural form of 'a'. Singular: Have you got a question? Plural: Have you got any questions? Singular: Have you got a programme? Plural: Have you got any programmes? You can only say 'Have you got any (+ singular noun) ? ' if that noun is uncountable, as in 'Have you got any water?' 'Have you got any money?' or if it's being used in an uncountable sense. 'Any' can also be used to mean 'it doesn't matter which'. When it is being used in that sense, it is possible to follow it with a singular countable noun. For example: Ask me any question you like. His parents let him watch any programme he wants.

Either a singular or a plural can follow any, depending on the context. Generally speaking (this is English - there are always exceptions to the rules!) if you were referring to lots of things you would use the plural - "any programmes". If you were referring to a single thing you would use the singular - "any programme". With your examples it would be normal to say "have you got any questions?". If I specifically wanted to restrict you to asking one question it would be gramatically correct for me to ask "have you got any question?" although in this case almost everybody would actually say "have you got a question?".