Learn Finnish: Onko kaupungin vanhoissa linnoissa valkoinen lintu? - Is there a white bird in the city's old castles?

When adding the plural -i- marker to a two-syllable word ending in -a (like vanha or linna), the -a changes to an -o- to make it flow better. Then you add the inessive ending -ssa (meaning "in"). This creates the plural stem vanhoi- + -ssa = vanhoissa.

If the castles are plural (linnoissa), why is the verb singular (onko)?

This is an "existential" sentence, which tells us that something exists in a certain location. In Finnish, these sentences always use the singular verb on (or the question form onko), no matter how many locations there are. The verb here is just establishing that the singular subject (lintu) exists.

Where does the g in kaupungin come from?

The basic word for "city" is kaupunki. When we put it in the genitive case to mean "the city's" by adding the -n ending, we close the final syllable. This triggers a consonant gradation rule where the strong nk softens to ng, turning kaupunki into kaupungin.

Why does the adjective have the plural "in" ending just like ?