Is there a difference grammatically in Hebrew when writing bottles of wine or wine bottles?
You can write בקבוקי יין or בקבוקים של יין, it means the same thing, similar to "wine bottles" and "bottles of wine".
Why isn't "How many wine bottles do you have?" accepted?
Don't know. it should be accepted
Maybe I'm missing something obvious, but why is there a י ať the end of בקבוק?
It's the masculine plural marker. You use it instead of "-ים" on the first word (or non-final word) in a construct state.
'Have you' is acceptable in English for 'do you have'