'szanować' takes Accusative, 'wujek' is surely masculine and animate, so the right form is 'mojego'.
Is it more common in Poland in sentences like this to use mojego or swojego?
Hard to say. With 1st person pronouns, both versions are natural. But perhaps it's safer to learn that if only a form of "swój" is correct, it's better to use that one.
is there always genetiv after szanuję?
No, it is Accusative Case after "szanuję", but accusative and genitive of "wujek" are the same ("wujka") https://pl.wiktionary.org/wiki/wujek
Szanuję swojego wujka.
Is the pronoun required here?
No, not really.