There seems to be no connection. The book ("liber") derives from what the inner bark of plants was called, while the liberty meaning ("līber") would be the same in latin (unrestricted, free, not slave).
The pronunciation differs between the two words, with a long i in the liberty meaning. I have very small knowledge of the pronunciation, and long/short i doesn't seem to be indicated in the Duolingo course.
Organic material like bark and leaves was easy to write on, so I guess the book meaning would have that connection, using bark to write on. There are some birch bark manuscripts preserved from the 1st century.
They're not, but it created many an amusing pun for speakers of Latin.
Liberty (freedom) from liber (free) is is from Indo-European hléwdʰeros (via Old Latin loeber, Proto-Italic louðeros), cognate with English 'lede', Russian 'ljudi' meaning 'people'. Thus to be a person or citizen was to be a free-man as opposed to a slave.
Liber (book) is from the inner bark of a tree, from Indo-European lubʰrós (via Proto-Italic luβros), from lewbʰ- (to cut off, peel), thus cognate with English 'leaf'.
So liber (free) is cognate with lede (people), and liber (book) is cognate with leaf (page). The problem is that Indo-European dʰ becomes Latin B, and this sound change is very strange.
Libri is the accusative case of libri. Libri is just the plural form of liber
Nominative = base form, to use when the word is the subject of the sentence, like "The books are there" (then you would use libri). You use the accusative case (libros) for the word that the subjuct acts on.