Learn Hardcore Greek: Ο ξάδερφός μου ερωτεύτηκε μια φοιτήτρια από άλλη χώρα. - My cousin male fell in love with a female student from another country.

Leaving out the article (ξάδερφός μου) is possible in some contexts, but the default, natural form in a sentence like this is with the article: Ο ξάδερφός μου.

Why does the accent move in ξάδερφος vs ξάδερφός μου?

On its own, the word is ξάδερφος (one accent, on the ξά).

When you add an unstressed possessive pronoun like μου, σου, του, Greek adds a second stress mark on the noun if the stress would otherwise be more than three syllables from the end of the whole phrase. This is a phonological rule.

Alone: ξά-δερ-φος → ξάδερφος
With μου: ξά-δερ-φος μου → written as ξάδερφός μου

You will see the same with:

ο φίλος → ο φίλος μου
ο άνθρωπος → ο άνθρωπός μου

The pronunciation gains an extra “bump” of stress before the enclitic μου, and the extra accent mark shows that.

What exactly is ερωτεύτηκε in terms of tense and person, and how is it translated?

ερωτεύτηκε is:

3rd person singular
aorist (simple past)
indicative
middle voice
of the verb ερωτεύομαι (to fall in love)

So ερωτεύτηκε means he fell in love (one completed event in the past).

Compare:

ερωτεύεται = he is falling in love / he falls in love (present)
ερωτευόταν = he was falling in love / used to fall in love (imperfect)
ερωτεύτηκε = he fell in love (aorist, simple past)

Why is the verb in the middle voice (ερωτεύτηκε, from ερωτεύομαι) and not an active form like ερωτεύσε?

In modern Greek, ερωτεύομαι (middle/passive form) is the normal, standard way to say to fall in love. The active ερωτεύω exists but is rarely used, and when it is, it usually means to make someone fall in love, which is different.

So:

ερωτεύτηκε μια φοιτήτρια = he fell in love with a student
(Hypothetical) ερωτεύσε μια φοιτήτρια would sound strange or would suggest he caused a student to fall in love.

Many Greek verbs for emotions or inner states are commonly used in the middle/passive:

φοβάμαι = I am afraid
θυμάμαι = I remember
ερωτεύομαι = I fall in love

Why doesn’t Greek use a preposition like “with” after ερωτεύτηκε? In English we say “fell in love with a student”.

In Greek, ερωτεύομαι κάποιον / κάποια takes a direct object in the accusative case, with no preposition:

ερωτεύτηκε μια φοιτήτρια = he fell in love (with) a female student
ερωτεύτηκε έναν φοιτητή = he fell in love (with) a male student

So the structure is: subject + ερωτεύτηκε + direct object, without a preposition like με.

What is the difference between μια and μία, and why is μια φοιτήτρια used here?

In practice, μια and μία are often pronounced the same in everyday speech ([miá] or a reduced [mja]), but there is a conventional distinction:

μια is usually the indefinite article = a / an
- μια φοιτήτρια = a (female) student
μία is usually the numeral = one
- μία φοιτήτρια = one student (emphasizing the number)

In many modern texts, you will see μια for the article, as in your sentence:
…ερωτεύτηκε μια φοιτήτρια… = …fell in love with a student… (not stressing “one”).

What exactly does φοιτήτρια mean, and how is it different from μαθήτρια?

φοιτήτρια means female university (or college) student.

φοιτητής = male university student
φοιτήτρια = female university student

μαθήτρια means female pupil / (usually) school student (primary or secondary school):

μαθητής = male pupil / (school) student
μαθήτρια = female pupil / (school) student

So:

μια φοιτήτρια = a (female) university student
μια μαθήτρια = a (female) schoolgirl / school student

Which case is μια φοιτήτρια, and how can I recognize it?

μια φοιτήτρια is in the accusative singular feminine.

Clues:

It’s the direct object of the verb ερωτεύτηκε, and direct objects are normally in the accusative.
The endings:
- Indefinite article (feminine, singular):
  - nominative: μια
  - accusative: μια (same form)
- Noun φοιτήτρια (feminine -α type)
  - nominative: η φοιτήτρια
  - accusative: τη φοιτήτρια

So the phrase μια φοιτήτρια (article + noun) is accusative here because of its function: it receives the action of the verb.

What does από άλλη χώρα mean, and why is από used instead of something else?

από άλλη χώρα means from another country.

από = from
άλλη = other / another (feminine, singular, accusative)
χώρα = country (feminine)

In Greek, από is the standard preposition to express origin or source:

από την Ελλάδα = from Greece
από το Λονδίνο = from London
από άλλη χώρα = from another country

You could also say μια φοιτήτρια άλλης χώρας (a student of another country), but that’s less neutral and slightly more formal. The everyday, natural way is από άλλη χώρα.

Why is it άλλη χώρα and not something like άλλο χώρα? How does άλλη agree with χώρα?

In Greek, adjectives must agree with the noun in gender, number, and case.

χώρα is feminine, singular, accusative in this sentence (because it’s the object of από).
So άλλος (other) must also be feminine, singular, accusative: άλλη.

Basic forms of άλλος:

Masculine: άλλος, άλλον
Feminine: άλλη, άλλη
Neuter: άλλο, άλλο

So:

άλλη χώρα = another country (feminine)
άλλος άνθρωπος = another person/man (masculine)
άλλο πράγμα = another thing (neuter)

Is the word order fixed, or can we rearrange the sentence Ο ξάδερφός μου ερωτεύτηκε μια φοιτήτρια από άλλη χώρα?

Greek word order is more flexible than English, because case endings show who does what to whom. Your original sentence:

Ο ξάδερφός μου ερωτεύτηκε μια φοιτήτρια από άλλη χώρα.

is perfectly natural. But you could also say, for example:

Μια φοιτήτρια από άλλη χώρα ερωτεύτηκε ο ξάδερφός μου.
(A student from another country my cousin fell in love with.)
– This puts emphasis on the student.
Ο ξάδερφός μου μια φοιτήτρια από άλλη χώρα ερωτεύτηκε.
– Emphasis shifted slightly to the object; sounds more marked/poetic.

The most neutral, everyday order for this meaning is the original:
Ο ξάδερφός μου ερωτεύτηκε μια φοιτήτρια από άλλη χώρα.