Learn Hardcore Japanese: watasi no ie ni tiisana neko ga iru.

の is the genitive (possessive) particle. It links two nouns to show possession or attribution. So 私の家 literally means “the house of me,” i.e. “my house.”

Why is に used after 家 instead of another particle like で or へ?

に marks the location where something exists or an action takes place. In existential sentences with いる/ある, you always use に to indicate “at” or “in” a place. で would emphasize the place of action (e.g. “I study at home”), and へ indicates direction (“to home”), neither of which fits “there is a cat at my house.”

Why is が used before いる instead of は?

In Japanese, existential sentences (those stating existence, like “there is/are…”) typically use が to mark the subject, because the focus is on the existence of something new or specific. は would shift the emphasis to a known topic (“As for cats, they exist at my house”), which sounds odd here if you’re simply stating that a cat exists at your place.

Why do we use いる instead of ある in this sentence?

いる is used for animate beings (people, animals) to express existence. ある is used for inanimate objects (books, cars, ideas). Since a cat is alive, we must use いる.

What’s the difference between 小さな and 小さい when saying “small”?

Both mean “small,” but 小さな is a prenominal adjective (used only directly before a noun), whereas 小さい is an い-adjective that can be used predicatively or prenominally. For example:

小さな猫がいる。 (Only before a noun.)
猫が小さい。 (As a predicate: “The cat is small.”)

Why does the verb いる come at the end of the sentence?

Japanese follows a Subject-Object-Verb (SOV) word order. Even in existential sentences, the structure is Location + Subject + Verb. Here:
私の家に (location)
小さな猫が (subject)
いる (verb).