Η ξαδέρφη μου γράφει βιογραφία για μια διάσημη γιαγιά από το χωριό της.

In Greek, nouns almost always take an article, even when they have a possessive like μου.
So:

• Η ξαδέρφη μου = my (female) cousin
  
  • Η: definite article, feminine, singular, nominative
  • ξαδέρφη: noun cousin (female)
  • μου: unstressed possessive pronoun my

Leaving out Η (Ξαδέρφη μου γράφει…) sounds incomplete or wrong in standard Greek.
English drops the article with possessives (my cousin), but Greek normally keeps it: η μητέρα μου, ο φίλος μου, το σπίτι μου, etc.

Greek possessive pronouns usually appear as enclitics (unstressed little words) after the noun:

• η ξαδέρφη μου = my cousin
• το βιβλίο σου = your book
• η μητέρα του = his mother

If you put the possessive before the noun, you must use the stressed form and it sounds emphatic or contrastive:

• η δική μου ξαδέρφη = my cousin (as opposed to someone else’s)

So the neutral, everyday way is [article] + [noun] + [clitic possessive].

Yes. Greek distinguishes gender for cousin:

• η ξαδέρφη = female cousin
• ο ξάδερφος = male cousin

So:

• Η ξαδέρφη μου γράφει… = My (female) cousin is writing…
• Ο ξάδερφός μου γράφει… = My (male) cousin is writing…

Notice in ο ξάδερφός μου that the article gets an extra accent (ξάδερφός) when a clitic like μου follows, to keep the stress on the third syllable from the end (a phonological rule).

Both forms exist and are understood:

• ξαδέρφη – more common in everyday modern speech and writing
• ξαδέλφη – slightly more formal / older spelling, still correct

They mean the same thing (female cousin). You’ll encounter both in texts; pick one style and use it consistently.

Yes. The Greek present tense γράφει covers both:

• She is writing a biography (right now / these days)
• She writes a biography (habitually, though that’s less natural in this exact sentence)

Greek present is used for both simple and progressive meanings in English.
If you wanted to clearly talk about a one-time, completed action in the past, you would use the aorist:

• Η ξαδέρφη μου έγραψε βιογραφία… = My cousin wrote a biography…

In this sentence, γράφει βιογραφία presents the activity in a more general or “type-of-work” way, a bit like English “she writes biography” (as an activity), though in English we normally say a biography.

All of these are grammatically possible, with slightly different nuances:

• γράφει βιογραφία – focuses on the activity of writing biography, a bit generic/stylistic
• γράφει μια βιογραφία – she is writing a biography (one, unspecified biography)
• γράφει τη βιογραφία – she is writing the biography (a specific, already known one)

In everyday speech, γράφει μια βιογραφία would probably be the most neutral way to say she is writing a biography (of someone).

βιογραφία primarily means biography, a life story.

For a CV/resumé, modern Greek usually uses:

• βιογραφικό or βιογραφικό σημείωμα

So in this sentence, βιογραφία is clearly a biographical work, not a job CV.

για is very flexible. Common meanings include for, about, because of. After verbs like γράφω, μιλάω, διαβάζω, it often means about:

• γράφω για κάποιον = I write about someone
• μιλάω για το βιβλίο = I talk about the book

So γράφει βιογραφία για μια διάσημη γιαγιά is best understood as:

• She is writing a biography *about a famous grandmother.*

Literally it uses για (for), but the natural English translation is about.

All three words agree in gender, number, and case:

• μια – feminine, singular, accusative, indefinite article
• διάσημη – feminine, singular, accusative, adjective famous
• γιαγιά – feminine, singular, accusative, noun grandmother

Greek requires this agreement:

• μια διάσημη γιαγιά = a famous grandmother
• μια διάσημη γυναίκα = a famous woman
• έναν διάσημο συγγραφέα (masc. acc.) = a famous writer

You can, but it sounds unusual or stylistically marked.
The normal, neutral order for an adjective is before the noun:

• διάσημη γιαγιά = standard, neutral: a famous grandmother

Putting the adjective after the noun (γιαγιά διάσημη) can sound:

• poetic
• emphatic
• or slightly old-fashioned / literary, depending on context

In everyday speech, stick with διάσημη γιαγιά.

Grammatically, της is just a feminine singular possessive clitic her.
By default, such a clitic usually refers to the subject of the sentence if nothing else forces a different reading, so many speakers will first interpret:

• από το χωριό της = from her (the cousin’s) village

However, context can easily make it refer to the grandmother instead. The pronoun itself is ambiguous.

To make it unambiguous, Greek would typically repeat the noun:

• από το χωριό της ξαδέρφης μου = from my cousin’s village
• από το χωριό της γιαγιάς = from the grandmother’s village

Χωριό is a neuter noun. The form το χωριό is neuter, singular, accusative:

• από το χωριό – from the village

Prepositions like από (from), σε (to / in), για (for/about), etc., take the accusative case in Greek, so:

• το χωριό (nom./acc.) – the village
• στο χωριό = to/in the village
• από το χωριό = from the village

Yes, but the meaning changes slightly:

• από το χωριό της = from her village (specific person’s village)
• από το χωριό = from the village (some previously mentioned or obvious village, but without specifying whose)

If the context already makes it clear which village you mean, από το χωριό is fine. If you want to emphasize that it’s her village, keep της.

Greek word order is more flexible than English because case endings show each word’s role. Variants like these are possible:

• Η ξαδέρφη μου γράφει μια βιογραφία για μια διάσημη γιαγιά από το χωριό της.
• Μια βιογραφία για μια διάσημη γιαγιά από το χωριό της γράφει η ξαδέρφη μου.

The basic, most neutral order is:

• [Subject] [Verb] [Object] [Prepositional phrases]
  → Η ξαδέρφη μου γράφει βιογραφία για μια διάσημη γιαγιά από το χωριό της.

Moving elements around usually adds emphasis or changes what is presented as “new” versus “given” information. For learning purposes, stick to the neutral order.