Almost. The dash (-), "Дима - медик," (Dima - medik.) is "Dmitrij jest medykiem," in Polish. The dash (-) is a substitution for the omitted present tense verb "to be/is" есть (jest'). It is not omitted in Polish. In Russian, it's left in only for emphasis. Дима есть медик! (Dima jest' medik!) For example, if you're arguing with someone about him being a medic, and they're denying that he's a medic.
"Dmitrij to medyk," has a direct translation in Russian "Дмитрий это медик." (Dmitrij eto medik.)
Zwierzę is the noun in the regular, nominative case, like you would look up in the dictionary. After the verb "to be" in Polish, być, in this case "is"/"jest", the noun declines into the instrumental case. That declension is zwierzęciem. It means that a cat is "in the state of being an animal."
If you omit the verb jest and say,
Kot to zwierzę.,
then zwierzę stays in the nominative case.
In this sentence you can use both. No difference in meaning ,
Noun is Noun - You can use "to", "to jest", "jest to" bot)h nouns in Nominative (easier) OR you can use "jest" - second noun (object) in Instrumental.
Pronoun is noun - You can use "to" "jest to" "to jest" with on/ona/ono/ one/oni, (he/she/it they) you cannot skip pronoun. (you cannot use it with I, you sing., we, you pl) Pronoun is noun- you can use different forms of "być= jestem/jesteś/jest/....) + noun in instrumental , you can skip pronoun.
Pronoun or noun is adjective - you don't use "TO" adjective in nominative