La meva germana vol plegar la roba abans de planxar-la.

In normal Catalan, possessives usually come with the definite article:

• la meva germana
• el meu llibre
• els nostres amics

That is different from English, which normally says just my sister without the.
Catalan does have some special or more old-fashioned/dialectal family forms like ma mare or mon pare, but la meva germana is the standard, neutral pattern.

Because the possessive agrees with the noun it describes, not with the speaker.

• germana is feminine singular, so: la meva germana
• germà is masculine singular, so: el meu germà
• plural feminine: les meves germanes

So meva matches germana.

Vol is the 3rd person singular present of voler: wants.

After voler, Catalan uses an infinitive directly:

• vol plegar
• vull menjar
• volen sortir

English uses to: wants to fold.
Catalan does not add a separate word equivalent to English to here.

Here, plegar means to fold or to put away clothes/laundry.

That verb can have other meanings in other contexts, such as:

• to stop work
• to leave
• to quit

So learners often get confused by it. In this sentence, because the object is la roba, the meaning is clearly the clothes/laundry sense.

Because roba is often a collective or mass noun in Catalan, like clothing or laundry in English.

So Catalan commonly says:

• la roba = clothes / clothing / laundry

Even though English often uses a plural word, Catalan uses a singular noun here.
That is also why the pronoun later is singular feminine: -la.

Because abans de is the normal pattern before an infinitive:

• abans de sortir = before leaving
• abans de menjar = before eating
• abans de planxar-la = before ironing it

A useful extra rule:

• abans de + infinitive when there is no new subject expressed
• abans que + subjunctive when a different subject is introduced

For example:

• Abans de marxar...
• Abans que ell marxi...

-la is a direct object pronoun meaning it.
It refers back to la roba.

So:

• planxar-la = to iron it

Because la roba is feminine singular, the pronoun is la.

With an infinitive in Catalan, unstressed object pronouns are normally attached to the end of the verb:

• planxar-la
• veure'l
• comprar-ne

This is called enclitic placement, and the hyphen is part of the normal spelling.

Before a conjugated verb, the pronoun usually goes before the verb:

• la planxa = she irons it
• la veu = he/she sees it

But with the infinitive, the normal form is planxar-la, not la planxar.

No. Each verb needs its own object.

• plegar la roba: la roba is the object of plegar
• planxar-la: -la is the object of planxar

English works similarly:

• fold the clothes before ironing them

So the pronoun is not unnecessary repetition; it is the object of the second verb.
You could sometimes omit it if the meaning were obvious from context, but keeping it makes the reference clear.