It is the singular feminine definite article in Spanish, "the" is the definite article in English. English uses only "the", but Spanish has four definite articles depending on noun gender and singular/plural.

Yes, "la" is the same as "the" in English. I would change the sentence, though, to "La esposa de Paco se llama Lucia"

"La" is a determinant for femenine singular, so you can use it for speak of a especific sustantive who you know. For example, " La prima de mi novio" but if you don't know who is the cousin of your boyfriends you talk, you shall use "una", "Una prima de mi novio". In other hand, there is a mistake in your sentences, you shall say " se llama" or "es llamada".

陰性單數定冠詞. 西班牙語名詞分陰陽性