Is there a Prohibition of not owning a non-kosher pet such as a puppy or cat etc.?

Regarding a dog specifically – please see Halacha2go.com article #88.

There is no halachic prohibition regarding owning other animals (except pigs and dogs), although one would need to learn the laws of caring for them (food, Shabbos etc.).

In general, we are discouraged from looking even at pictures of non-kosher animals, as the negative traits they posses have an effect on us – especially children.

For more on this topic see:

The Rebbe’s Sicha:

(Yiddish) Likkutei Sichos vol. 25 pg 309.
(Lashon Hakosdesh) Toras Menachem 5744 vol. 1 pg. 487.
(English) here