Learn Finnish: Tykkäätkö enemmän sekarotuisesta vai puhdasrotuisesta kissasta? - Do you prefer a mixed-breed or a purebred cat?

They are compound property adjectives. You take a modifier, like seka- (mixed) or puhdas (pure), attach the noun stem rotu (breed/race), and add the adjective suffix -inen. This pattern creates an adjective describing a characteristic, meaning "having a mixed breed" or "having a pure breed".

Why do both of these adjectives and the word for cat end in -sta?

The verb tykätä (to like) requires the thing being liked to be in the elative case (-sta / -stä). Because the main noun is kissasta, its modifiers must agree with it. When words ending in -inen are put into the elative case, the ending changes to -isesta (e.g., sekarotuisesta, puhdasrotuisesta).

There is no direct translation of the verb "prefer" here. How does the sentence convey that meaning?

In Finnish, "to prefer" is most commonly expressed by combining the verb tykätä (to like) or pitää (to like) with the comparative adverb enemmän (more). Literally, the sentence is asking "Do you like a mixed-breed or a purebred cat more?"

Why does the sentence use vai for "or" instead of tai?