Hún býr í kjallaranum.

býr is the third person singular present indicative of the verb búa, which means “to live” or “to reside.”
Conjugation of búa in the present tense:

• ég bý (I live)
• þú býrð (you live)
• hann/hún/það býr (he/she/it lives)
• við búum (we live)
• þið búið (you pl. live)
• þeir/þær/þau búa (they live)

The preposition í when expressing location (where someone is) governs the dative case in Icelandic. Since kjallari (basement) is the object of í, it must appear in the dative:

• nominative: kjallari
• dative: kjallaranum

Sometimes you can, because Icelandic verbs clearly mark person and number. However,

• “býr” alone could be hann, hún or það.
• Omitting Hún makes the sentence ambiguous: Býr í kjallaranum could mean “He/She/It lives in the basement.”
• Use Hún when you need to specify that it is “she.”

Yes. Icelandic is a V2 language (verb-second). You could front the locative phrase to emphasize place:

• Í kjallaranum býr hún. (“In the basement she lives.”)
  This stresses where she lives.

Phonetic approximation in English: “CHYAT-lah-RAH-num”

• kj = a soft “ch(y)” sound (like ji in “jeep,” but palatalized)
• ll = a voiceless lateral fricative (similar to Welsh “ll”)
• Stress on the second syllable: kjall​a​RA​num

kjallari is a masculine noun. Clues:

• It ends in -i in the nominative singular, a common masculine pattern.
• In dictionaries you’ll see it marked “kk” (karlkyn).
• Its article/ending patterns (kk: kjallari, kjallarans, kjallaranum, kjallari) match masculine declension.