Thank you for your replies; they're very helpful
Yes, I'm a native speaker of English (Can't always rest on your NS English laurels, when you're teaching the grammar, I'm afraid, for me at least.)
How about this sentence?:
-'If you can have dinner with a famous person, who will you choose?'
It's unusual, and there'd be almost no context for it, but it is grammatical in the right context. Yes, it's first conditional.
This sentence is in the 1st conditional but it doesn't 'sit right' for me. 'Having dinner with a famous person' is a distinctly fantastic/ unreal experience, so I'm guessing for that reason the sentence must be rendered in the 2nd conditional
-'If you could have dinner with a famous person, who would you choose?'
This is in the second conditional; it would be far more common. But it's a different sentence with a different meaning from the first.
To your knowledge, is the 1st conditional in the above example ungrammatical given the context?
We don't have a context. We only have your sentences.
This is what led me to question if modal aux. verbs are possible in the 'if-clause' of the 1st conditional in the first place.
Oh, I thought you were asking whether "If I can get a haircut this weekend, I'll go to the party" was grammatical or not. Did you really doubt this?
Thank you for any feedback.