Breakdown of Podoba mi się twój kotek i domek.
i
and
twój
your
podobać się
to appeal / to like (aesthetically, first impressions)
mi
to me
Questions & Answers about Podoba mi się twój kotek i domek.
How do we form the diminutives kotek and domek?
Both are masculine nouns, so they use the suffix -ek. You take the base words kot (cat) and dom (house) and add -ek to make them small or affectionate.
Since I like two things, shouldn't the verb be plural (podobają)?
It can be, but it doesn't have to be! In Polish, when the verb comes before a list of subjects, it is very natural for it to just agree with the first one. Since kotek is singular, the singular verb podoba sounds perfectly normal here. Using the plural podobają mi się would also be correct.
Why are kotek and domek in the basic dictionary form (Nominative case) instead of an object case?
Because the verb podobać się literally means to appeal. The Polish sentence is built as: your kitty and little house appeal to me. The things you like are actually the subjects of the sentence, so they stay in the Nominative case!
Why does the sentence use twój (singular) instead of a plural word for your?
The word twój modifies kotek, which is a singular masculine noun. In Polish, possessive words usually just match the noun right next to them. It is understood that it applies to the domek as well, so you don't need to repeat it.