A la biblioteca, la meva cosina buscava un conte, mentre jo llegia en silenci.

Yes. In A la biblioteca, the preposition a means to or at, depending on context. Here it means at/in the library.

• a = to / at
• la biblioteca = the library

Catalan often uses a where English might use in for locations, especially with places. So A la biblioteca is a very natural way to say At the library.

Catalan often uses the definite article before a possessive:

• la meva cosina = my cousin
• literally: the my cousin

This is normal in Catalan and does not sound strange to native speakers. English usually does not do this, but Catalan often does with family members, body parts, and many everyday nouns.

So la meva cosina is the standard form here.

Yes. Cosina is female cousin.

• el cosí = male cousin
• la cosina = female cousin

Since the sentence says la meva cosina, we know the cousin is female.

Buscava is the imperfect form of buscar (to look for).

Here it shows an action that was ongoing in the past:

• buscava = was looking for

So la meva cosina buscava un conte means my cousin was looking for a story/book.

Catalan uses the imperfect, like Spanish and other Romance languages, for background or continuing past actions. In this sentence, both actions were in progress at the same time:

• she was looking
• I was reading

That is why buscava fits well.

Even though it may look similar to an English word, conte in Catalan means story, tale, or sometimes short story.

Examples:

• un conte infantil = a children's story
• un conte de fades = a fairy tale

So in this sentence, buscava un conte means was looking for a story / a tale / a book of stories, depending on context.

Mentre means while.

It connects two actions happening at the same time:

• la meva cosina buscava un conte
• mentre jo llegia en silenci

So the sentence means:

• My cousin was looking for a story, while I was reading in silence.

It is a very common word for linking simultaneous actions.

Yes, Catalan could leave it out, because the verb form already shows the subject:

• llegia already tells us it means I was reading

So mentre llegia en silenci could also work in many contexts.

However, jo is often added for:

• emphasis
• contrast
• clarity

Here it helps create a contrast:

• my cousin was doing one thing
• I was doing another

So mentre jo llegia en silenci feels natural and slightly more contrastive.

Llegia is the imperfect of llegir (to read).

It means:

• I was reading
• I used to read in other contexts

In this sentence, it clearly means I was reading, because it describes an action in progress in the past, happening at the same time as buscava.

So both verbs are in the imperfect because both actions form the background scene.

Yes, llegir has some stem changes in certain forms, but llegia is a normal imperfect form.

The infinitive is:

• llegir = to read

The imperfect forms include:

• jo llegia
• tu llegies
• ell/ella llegia
• nosaltres llegíem
• vosaltres llegíeu
• ells/elles llegien

So llegia is simply the I/he/she imperfect form.

En silenci means in silence or silently.

This is a fixed and very common expression:

• parlar en veu baixa = to speak quietly / in a low voice
• treballar en silenci = to work in silence
• llegir en silenci = to read in silence

English often uses an adverb like silently, but Catalan frequently uses a prepositional phrase:

• en silenci = in silence

The commas help separate parts of the sentence clearly:

• A la biblioteca, sets the scene
• then comes the first action
• then mentre introduces the second action

So the commas make the structure easier to follow:

1. location
2. first ongoing action
3. second ongoing action

They are natural punctuation here, especially because the opening phrase A la biblioteca is placed before the main part of the sentence.

Yes. Catalan word order is flexible, though some versions are more natural than others.

For example, these are possible:

• La meva cosina buscava un conte a la biblioteca, mentre jo llegia en silenci.
• Mentre jo llegia en silenci, la meva cosina buscava un conte a la biblioteca.

The original sentence starts with A la biblioteca to set the scene first. That is a very common and natural choice.

A fairly literal version would be:

• At the library, my cousin was looking for a story, while I was reading in silence.

This helps show the structure of the Catalan sentence more closely than a freer English translation might.