When is "Does... have...?" correct versus "Is... having... ?"

I have to generate random questions Yes/No about hotels, restaurants, etc. for a Natural-Language Programming task. The focus is on questions about characteristics about such places that are rather dynamic (e.g., current length of queue, available parking spaces).

I'm trying to cover a wide range of formulations so that not all questions look too much alike (e.g. Is KFC nice? Are the rooms in Hilton Hotel large?). As a non-native English speaker, I stumble upon some problems to guarantee grammatically correct questions. Given the following two example questions:

• Does [RESTAURANT-NAME] have a promotion?
• Is [RESTAURANT-NAME] having a promotion?

I would say that both questions convey the same meaning, with the latter maybe emphasizing the current moment (implying that promotions are rather infrequent and dynamic). Is this correct?

My follow-up question is now: When I can use both forms interchangeably? For example, when I have

• Does [RESTAURANT-NAME] have vegan dishes?
• Is [RESTAURANT-NAME] having vegan dishes?

The second one "feels" wrong since the menu usually doesn't change much over time. Am I correct to say that the possibility to formulate a "Is/Are... having... ?" depends on the meaning/semantics of the question? In other words, I cannot always use both formulations and I cannot trivially decide in a program?

The word have has multiple definitions. Here are two:

have verb

1 Possess, own, or hold. ‘he had a new car and a boat’

4 Perform the action indicated by the noun specified (used especially in spoken English as an alternative to a more specific verb) ‘We will be having a meeting soon to examine our options, to see what is possible.’

- ODO

Your promotion example uses definition 4 (action) whereas your vegan dishes example uses definition 1 (possession).

You ask:

Am I correct to say that the possibility to formulate a "Is/Are... having... ?" depends on the meaning/semantics of the question? In other words, I cannot always use both formulations and I cannot trivially decide in a program?

Yes. When used in the sense of possession, the is having form sounds awkward.

• We do NOT normally use stative verbs in the continuous (‘ing’) form - Stuart Cook, What are stative verbs and why do learners need to understand them?

• Stative verbs usually don’t use present progressive (or other progressive tenses). - page 34, Kent Uchiyama, English Verb Tenses

• Some verbs describe a state or condition that does not change (e.g., seem, appear, feel, own, prefer, doubt, recognise). These stative verbs are special in that the continuous form of the verb is not possible. - Better Grammar, Curtin University

In your first example, to clearly ask whether or not the restaurant currently has a promotion, I would use the progressive tense, since (as you said) it conveys the notion that promotions are rare. The simple present tense isn't that incorrect, but it sounds more awkward.

Is [Restaurant] having a promotion?

In your second example, I would use the simple present tense, since (as you said) the dietary content of the menu (whether or not they accommodate vegans) shouldn't change with the specific dishes. Although, if you wanted to make it sound better, you might use serve instead of have.

Does [Restaurant] serve vegan dishes?