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 ...
I'm only at an intermediate level of Latin, but I thought I'd make an attempt to express your request. Maybe there's something here you can use—or an error here that someone will catch.
Here's a more prosaic version:
Lectiones praescriptas legere potui ultraque cupio. Dabisne mihi plures lectiones itaque plura puncta meream?
And here's a more histrionic ...
As a general rule, Latin verbs go at the end of the sentence, including the verb sum. (Traditionally, Latin has been considered a "SOV" (subject, object, verb) language, though scholars have recently cast doubt upon the necessity of that claim.)
Having said that, verb placement can vary freely, mostly for pragmatic reasons---say if you want to emphasise ...