Learn Polish: Wbrew pozorom to bardzo spokojny owad. - Contrary to appearances, it is a very calm insect.

The preposition wbrew meaning contrary to or against always requires the Dative case. Here it takes the plural noun pozory (appearances) and changes it to the Dative plural form pozorom.

Is wbrew pozorom a set phrase in Polish?

Yes, it is a very common fixed idiom meaning contrary to appearances or despite what it looks like. However, you can also use wbrew with other nouns, such as wbrew prawu (against the law) or wbrew mojej woli (against my will), always remembering to apply the Dative case to the noun that follows.

Why does the sentence use to instead of jest to mean it is?

In Polish, when you equate a subject with a noun phrase, you often use the particle to followed by the Nominative case (here, bardzo spokojny owad). If you wanted to use the verb jest instead, the noun phrase would have to switch to the Instrumental case: on jest bardzo spokojnym owadem.

Does the fact that owad is an animate noun change how we treat spokojny here?