What is the difference between "psy" and "psi"? This example uses first variant but in some others "psy" is marked wrong.
PSI is nominativ. PSY is acusativ and instrumental.
Is the extra tvé necessary? In English 'I hear your dogs and cats' would be more common I think. An extra 'your' to me seems like excessive emphasis
As a stand-alone English sentence, "I hear your dogs and cats" is fine. But the second "your" should be in the translation here because it appears in the Czech sentence.
Is the extra tvé necessary in the Czech sentence grammatically speaking?
No, it is not grammatically necessary in either language. It changes the sentence somewhat so it should be kept in the translation.