Η μαθήτρια δεν βρίσκει τη γόμα της, οπότε της δίνω τη δική μου ξύστρα και ένα άλλο μολύβι.

They are the same form, but they do two different jobs:

• τη γόμα της = her eraser
  Here της is a possessive pronoun, meaning her.

• της δίνω = I give her
  Here της is an indirect object pronoun, meaning to her / her.

So in one place της shows possession, and in the other it shows the person receiving something.

In Greek, possession is often shown with a weak pronoun after the noun:

• η γόμα = the eraser
• η γόμα της = her eraser

Literally, this is closer to the eraser of hers, though in natural English we just say her eraser.

This is a very common Greek pattern:

• το βιβλίο μου = my book
• ο φίλος σου = your friend
• η τσάντα της = her bag

Because γόμα is the object of the verb βρίσκει.

• η γόμα is nominative: used for the subject
• τη γόμα is accusative: used for the direct object

Here, the student is doing the action, and the eraser is what she cannot find:

• Η μαθήτρια = subject
• τη γόμα = direct object

So Greek changes the article to show case.

Βρίσκει is the 3rd person singular present form of βρίσκω = find.

So:

• βρίσκω = I find
• βρίσκεις = you find
• βρίσκει = he/she/it finds

Since η μαθήτρια is she / the female student, Greek uses βρίσκει.

Also, Greek present tense often covers both:

• she does not find
• she is not finding

depending on context.

In standard Greek, δεν is the basic word for not with indicative verbs.

So:

• βρίσκει = she finds
• δεν βρίσκει = she does not find

This is the normal way to negate a statement in the present tense.

Οπότε here means something like:

• so
• therefore
• as a result

It links the two parts of the sentence:

• The student can’t find her eraser, so I give her...

It is a very natural connector in everyday Greek.

Greek often uses a weak pronoun without a separate word for to when English would say to her / him / them.

So:

• δίνω στη μαθήτρια = I give to the student
• της δίνω = I give her

Both are possible, but της δίνω is shorter and very natural.

Here της is in the genitive form, which is commonly used for indirect objects in Modern Greek pronouns.

Because Greek does not normally say τη μου ξύστρα.

To say my own sharpener or to stress possession, Greek uses:

• τη δική μου ξύστρα

The word δική comes from δικός / δική / δικό, which means one’s own.

So:

• τη δική μου ξύστρα = my own sharpener / mine

This structure is especially common when:

• you want to emphasize whose thing it is
• the possessed item is being contrasted with someone else’s

In this sentence, it sounds natural because the speaker is giving their own sharpener.

Yes, but the meaning would be slightly different.

• της δίνω μια ξύστρα = I give her a sharpener
• της δίνω τη δική μου ξύστρα = I give her my own sharpener

So the original sentence emphasizes that the speaker lends or gives the sharpener that belongs to them.

Άλλο means another or other / different, depending on context.

So:

• ένα μολύβι = a pencil
• ένα άλλο μολύβι = another pencil / a different pencil

Here it means the speaker also gives the student one more pencil, probably a replacement.

Because μολύβι is neuter singular, the forms are:

• ένα = a/an
• άλλο = other/another (neuter singular)

Because they are different kinds of noun phrases:

• τη δική μου ξύστρα = my sharpener
  This is a specific item, so Greek uses the definite article τη.

• ένα άλλο μολύβι = another pencil / a different pencil
  This is an indefinite item, so Greek uses ένα = a/an.

Greek uses articles a lot, and this difference is very normal:

• definite = the / this specific one
• indefinite = a / another

This sentence is a good example of several important Greek case uses:

• Η μαθήτρια — nominative
  The subject of the sentence.

• τη γόμα, τη δική μου ξύστρα, ένα άλλο μολύβι — accusative
  Direct objects.

• της in τη γόμα της and της δίνω — genitive form
  Used here for possession and for the indirect object pronoun.

So even in one short sentence, Greek shows several relationships through case forms.

Yes, Greek word order is more flexible than English because case and pronouns show the grammatical roles.

The given order is very natural:

• της δίνω τη δική μου ξύστρα = I give her my sharpener

But Greek can move things for emphasis, for example:

• τη δική μου ξύστρα της δίνω
  This sounds more emphatic, like It’s my sharpener that I give her.

The version in your sentence is the most neutral and natural everyday order.

Because μαθήτρια means female student / schoolgirl, while μαθητής means male student / schoolboy.

So:

• ο μαθητής = the male student
• η μαθήτρια = the female student

Since the sentence refers to a female student, Greek uses μαθήτρια.

In this context, ξύστρα means pencil sharpener.

Greek words can sometimes have broader or different uses in other contexts, but here, because the sentence also mentions μολύβι and γόμα, it clearly belongs to school-supplies vocabulary:

• γόμα = eraser
• ξύστρα = sharpener
• μολύβι = pencil

So the meaning is completely natural here.