Russian usually omits the verb 'to be' in Present tense, English very rarely does. It makes little sense to translate a normal and natural Russian sentence with a not very natural English sentence.
Russian punctuation rules requires a dash here, regardless of whether you use «это».
I think it's important to make the distinction early to someone learning Russian fresh with DL. Also, the sentence in Russian is a complete sentence whereas Tom, my favorite cat (which was my sentence and I also omitted to be in Eng) is not a complete sentence. It could qualify as the subject of sentence followed by an appositive or qualifying statement.
It doesn't seem to me that it matters for this question: if the pets are persons, you have specified your favourite person of a particular species, who may be different from your favourite person overall; if not, then your favourite dog and cat have nothing to do with your favourite person. You can still say [Наташа] - мой любимий человек.
LOL, no, but I did study. "Eres un abogado?" would be good (ser and estar have an slightly different meaning in spanish). I pointed the use of "person" for animals is deeply related with the idea of them having rights and obligations. I understand thats the difference between being a "being" and a "person".
Кот is the masculine, кошка is feminine.
Suffix -к- is indeed often used for creating diminutive forms, but sometimes it's used for feminine forms with no diminutive meaning (like in кошка).
Diminutives would be ко́тик and ко́шечка.
The usual word for 'kitten' is котёнок.
As for ки́ска, while it used to refer to cats, I wouldn't use it in modern Russian. It underwent a meaning change, like the English word '❤❤❤❤❤', so nowadays it's not often used to talk about cats, except in double entendres.