The word for "literature" is literally "letters", so the Latin word for "literature" is always in the plural.
Litteris is the plural dative of littera. The verb "studere" takes the dative because it literally means "to devote oneself to".
Litteras is the plural accusative of littera. The verb "legere" takes the accusative because, just like in English, the literature is the direct object of the verb.
Well, believe it or not :), in Classical Latin the nominative was the same as vocative, i.e., a short /-a/ sound. The locative (place where), instrumental (with what), ablative (from where) merged (into Classical Latin ablative), which was a long /-ā/ [ah] (from earlier /-ād/). For example /schola/ 'school', but /in scholā/ 'in or at (the) school'. Since these both became just /-a/ in later Vulgar Latin and by extension the Romance languages, they were pronounced the same by that community. The special locatives such as Romae "in Rome" are archaisms preserved because of their frequency and reflect the fact that locative was earlier /-i/, i.e., Rōmae < /Rōm-a-i/.
Fun fact, just in case you are wondering why I said nominative was the same as vocative in this declension and not the other way round is because the nominative was originally a long /-ā/ (different from ablative which was then /-ād/ as stated earlier). The Classical Latin short /-a/ in the nominative is a puzzle, unless it was actually taken from the vocative which was a short /-a/ from the Indo-European times, which are by definition pre-Latin.
PS. I happen to agree with everyone here on the pronunciation: it is pitiful but so what?! Just get over it, I did. The male voice is closer to the restored (or classical) pronunciation, the female voice is more ecclesiastical or medieval. The course is fun regardless.
The default position of the verb is at the end. To put it elsewhere is to put a stress or nuance to the sentence.
sunt multae linguae would, to me, be emphasising the multiplicity of languages; both sentences valid, but with a slightly different purpose - and therefore delivery - to each.
It's generally not a good idea to split a single noun phrase with a verb (or at all, really). It changes the meaning by making it no longer a single noun phrase but rather two noun phrases being equated or compared.
Multae linguae sunt = There are many languages.
Multae sunt linguae = Many [of some unspecified thing] are languages.