Your analysis is correct: this is fīlia "daughter" + -ne "?".
The trick is, -ne can attach to any word, not just verbs. In fact, it usually attaches to either the first word, or the most emphatic word, whatever that might be. Since nōn often comes at the beginning, nōn-ne became common enough that you'll often see it analyzed as a word of its own, rather ...
This answer only considers the nuances of habere, not a comparison between it and the possessive dative.
The possessive genitive is different; it functions mostly like the English genitive and is used to express things like "my dog" rather than "I have a dog".
The example of the pope actually makes a good example for habere.
The canonical announcement upon ...
Allen & Greenough §343 also lists it as a type of possessive genitive, giving a few examples. Note that this use of the genitive in the predicate is used with infinitives and with clauses:
c. An infinitive or a clause, when used as a noun, is often limited by a genitive in the predicate:—
neque suī iūdicī [erat] discernere (B. C. 1.35), ...
I would take cujuscunque rei causa to be a single noun phrase: "a cause for/of every individual thing". That is, assignari has a subject, but the object is left implied; if it were made explicit, it would be something like illi "…to that thing".
A cause of every individual thing must be assigned [to that thing]: either a reason why it exists, or a reason ...
The difference is not great between "The cause and reason ought to be assigned to each thing why or why not..." and "The cause and reason of each thing why or why not... ought to be assigned..." Though the Genitive after causa is more literal.
I think the most significant reason for choosing the Genitive rather than the dative is continuity. The formula "...
Is this predicative genitive + esse?
"est boni imperatoris bene ducere" = "It is of a good general to lead well.";
"est digni civis patriae cogitare" = "it is of a worthy citizen to think about his country.";
"est honesti regis regere sapienter" = "It is of an honourable king to rule wisely.";
"est canis feles agitare" = "It is of a dog to chase cats."