Learn Icelandic: Er kærastinn þinn frá lítilli eyju? - Is your boyfriend from a small island?

The adjective has to match the noun it describes, which is eyju (island). Because eyju is a feminine noun in the dative case, the adjective must take the feminine dative strong ending. While this lesson teaches that the feminine dative ending is usually "-ri", for adjectives ending in "-l" like lítill, that ending blends into an "-li" sound, making it lítilli.

Why does the word for island end in a "u" (eyju) in this sentence?

The preposition frá (from) always forces the noun that follows it into the dative case. The basic dictionary form of the word is eyja, which is a feminine noun. Most feminine nouns ending in "-a" change their ending to "-u" in the dative case, so it becomes eyju.

Why is the definite article attached to kærastinn when the English translation just says "your boyfriend"?

This is a standard pattern in Icelandic. When you use a possessive pronoun like þinn (your), the noun it belongs to almost always takes the attached definite article (the "-nn" part of kærastinn). You are literally saying "the boyfriend your".

What case is kærastinn þinn in?

It is in the nominative case. It is the subject of the sentence, and the verb vera (to be) simply connects the subject to the rest of the sentence without changing its case.