Livia librum lego - I, Livia, read a book; Livia librum legis - You, Livia (not vocative), read a book; Livia librum legit - Livia reads a book
LIVIALIBRVMLEGO is even more confusing, but that’s what they would have written for this sentence in the first century BC.
The same, until Late Latin, when they may - if they wished - have added illum (literally “that”).
Yes. In Classical Latin it would mean “Livia reads that book”, but later “the book”.
Legunt is the third person plural form, in other words the form that is used for they or groups of people like Marcus et Livia librum legunt -> "Marcus and Livia read a book."
Legit is used since it is the third person singular form which is used for single individuals, like Livia by herself here.