Puella cum avia libentius sedet quam sola in cubiculo manet.

Because cum meaning with takes the ablative. So cum avia means with grandmother / with her grandmother.

A useful detail: avia is a first-declension noun, and in the singular its nominative and ablative forms both happen to be avia, so the form looks the same even though the function is different.

With nouns, cum normally comes before the noun: cum avia, cum amico, cum puella.

With personal pronouns, however, Latin usually attaches it to the end:

• mecum = with me
• tecum = with you
• nobiscum = with us
• vobiscum = with you all
• secum = with himself/herself/themselves

So cum avia is exactly what you should expect.

Libentius is the comparative adverb of libenter.

• libenter = willingly, gladly
• libentius = more willingly, more gladly, rather

Since the sentence is making a comparison with quam, Latin uses the comparative form. The sense is that the girl does one thing more gladly than the other.

Quam means than after a comparative word such as libentius.

Here it introduces the second half of the comparison:

• cum avia libentius sedet
• quam sola in cubiculo manet

So the sentence compares two actions/situations:

1. sitting with grandmother
2. staying alone in the room

Sola agrees with puella.

It is:

• feminine
• singular
• nominative

Even though puella is not repeated in the second part, the adjective still refers to the same subject. So the sense is the girl remains alone, not that some separate feminine noun is being introduced.

Because Latin often leaves out a repeated subject when it is already clear from the context.

Once puella has been stated, Latin does not need to say it again if the reader can understand that the same person is still the subject. The combination of context and sola makes that clear here.

This is very normal Latin style.

Because in takes:

• the ablative for location: in the room
• the accusative for motion toward: into the room

Here the girl is already there, or at least the phrase describes where she remains, so Latin uses the ablative:

• in cubiculo = in the room

If the idea were movement into the room, you would expect in cubiculum.

The present tense here can describe:

• a general truth
• a habitual action
• a scene happening now

Latin present tense often works much like English present tense, though English may sometimes translate more naturally with is sitting or stays depending on context.

So sedet and manet do not have to mean only one specific moment; they can also describe what she generally does or prefers to do.

They are not the same verb.

• sedet comes from sedere = to sit
• manet comes from manere = to remain, stay

So the first clause focuses on sitting with grandmother, while the second focuses on remaining alone in the room. Latin is being quite precise about the two different situations.

Yes, but in a normal Latin way. Latin word order is much more flexible than English word order, because the endings carry a lot of grammatical information.

This sentence is arranged clearly and naturally:

• Puella comes first, introducing the subject.
• cum avia stays together as a phrase.
• libentius is placed near sedet, the verb it modifies.
• sola comes before in cubiculo manet, which highlights the idea of being alone.

So the order is not random, but it is also not as rigid as English.

Often, yes. That is a very natural English way to express the idea.

Literally, libentius means more willingly / more gladly, but in smooth English the comparison can sound like would rather.

So Latin is not using a special separate verb meaning prefer here; instead, it uses a comparative adverb plus quam to express preference between two actions.

Because Latin has no articles.

Latin does not have separate words for the or a/an. Whether a noun should be understood as definite or indefinite depends on context.

So:

• puella can mean the girl or a girl
• avia can mean the grandmother / her grandmother / a grandmother
• cubiculo can mean the room or a room

English has to choose an article, but Latin usually does not state one.