cf. (1) "comparing the foods of A and B" (2) "comparing the food of A and B."

In case (1) we are explicitly comparing distinct classes or preparations of food (meals), while in case (2) we are comparing food (cuisine) as a whole. Thus I would say that they are equally valid, but may have slight contrast in meaning.

Sure, you usually discuss syntax as a whole, but you can also have subsets or specific cases of syntax which would be "syntaxes".

Syntax is synonymous with "sentence structure". Can you have plural "sentence structures"? Yes, you can. Thus also "syntaxes".

The general answer was covered above: both are possible, both may have slightly different uses, but they're effectively interchangable. (The only one that is required is the singular if two owners share a single thing-- the "relationship between English and French" for example.)

But in this case of "syntax", it's very hard to me to imagine a plural usage of that word. It is, I think, a mass noun, like "knowledge".
???The knowledges of the man and the woman.
?The syntaxes of English and French.

More than that, I just don't know what the form would be. According to several results online it is indeed "syntaxes", but that looks weird to me!

In this specific case, I am also positive I have seen that exact construction (syntax of X and Y) used in published work, so I'm sure that's fine.

It can, but it's unusual. The problem here is "of a pair of languages." It makes it seem as if by being in a pair, they are of the same sort of category. If you changed it to "a few" or "a group of" then you can get rid of the weirdness:

A comparative study of the syntax of a group of languages.

And no, I wouldn't say syntax being singular has anything to do with the singular nature of group.

But what if the sentence is like this?
"A comparative study of the syntaxes of a pair of languages ----"
can this be substituted by
" A comparative study of the syntaxof a pair of languages -----"

I told you above: grammatically in terms of the plural usage either could work.
But with the specific word "syntax(es)" I absolutely reject the plural usage. It screams "non-native usage!" or "weird linguistics terminology!"