"Where is my cat? I do not see it."
Translation:Где моя кошка? Я её не вижу.
No. That would be a literal translation, but no native English speaker would ever render the sentence this way in this context. Outside of a few highly specific exceptions, gendered pronouns are only used to refer to people. For animals, gendered pronouns are only used when talking about a specific animal of the corresponding gender. Some speakers personify animals to which they are closely attached and thus may use gendered pronouns with e.g. the family pet. However, this is not a rule, and it is never wrong to use the neuter "it" regardless of whether or not you have such specific knowledge.
Since we do not have such specific knowledge in this case, the English pronoun used must agree in gender with the English word "cat", which is neuter.
Afaik the words change position but the grammar structure is still the same. I guess it is correct but some other word shuffles would be faux pas and considered rude by native speakers even though still grammatically correct e.g. дайте мне кофе (normal) and кофе дайте мне (rude).
kot is a male cat and koshka is a female cat. you can use either one, but then you need to adjust all the other words to your choice. моя is for my-female-something. мой is for my-male-something. её is her and его is him so you either go with "Где моя кошка? Я её не вижу" or "где мой кот? я его не вижу"