The word "does" is not translated into or from Latin here - habitat means "he/she/it lives" - it does not mean "he/she/it does live."
Additionally: the "does" in your answer is used in modern English only for emphasis - to confirm that Corinna lives alone. If the sentence is on its own (without context), then "Corinna lives alone" is the correct sentence.
Not in the Latin course, apparently. (And since the alternative spellings would have to be added manually for each sentence, I'd rather that time be spent on adding alternate word orders, etc.)
According to the "tips" for the introduction "skill", https://www.duolingo.com/skill/la/Introduction/tips-and-notes
Translation of Names
A little convention: we will not accept translations of names as alternatives in this course. Marcus's name is Marcus, not Mark, and Stephanus is not Stephen or Steven.
Not necessarily the same ending. There are different declensions. Every noun has a gender, Corinna is feminine hence "sola". But "urbs" (city) is also feminine, does not end in "a" but an adjective like "solus-sola-solum" would also take feminine endings (i.e. "sola" in this case) when used to describe an "urbs".
há.bi.tō, -tās, -tat, (ha.bi.táà.mus, -tis), há.bi.tant (= just like in Italian).
This is because the syllable .bi. contains a short vowel, since it is a derivation based on the perfect passive participle of habeo: há.bi.tum < earlier /hab-ĕ-to-m/. There are some II. conjugation adverbs (ablatives) based on the supine participles with stems in this (short) -e- attested in Old Latin (like MER-E-TO-D "deservedly", cf. méritus of 'mereri' "to be deserving, i.e., to deserve", etc.).
As the Latin proverb says:
"Donec eris felix, multos numerábilis amicos.
(While you are wealthy, you will have a lot of friends)
Tempora si fuierint nubina, solus eris".
(But if times come hard for you, you will be as alone as a leper)
N. B.: Free translation into English
Tempora nubila = (black) cloudy weather.