Μετά την πτώση, το γόνατό της είναι λίγο πρησμένο και χρειάζεται ξεκούραση.

Because μετά here means after and it is followed by the accusative case.

• η πτώση = the fall (nominative)
• την πτώση = the fall (accusative)

So after μετά, Greek uses:

• μετά την πτώση = after the fall

This is a very common pattern:

• μετά το μάθημα = after the lesson
• μετά τη δουλειά = after work

Yes. Μετά την πτώση and μετά από την πτώση are both possible.

In everyday Modern Greek, both are used to mean after the fall. Very roughly:

• μετά την πτώση = slightly more compact/direct
• μετά από την πτώση = also very common, often a bit more conversational or explicit

So this sentence could also be:

• Μετά από την πτώση, το γόνατό της είναι λίγο πρησμένο και χρειάζεται ξεκούραση.

Both sound natural.

Της here is the weak possessive form meaning her.

Greek often expresses possession like this:

• το γόνατο = the knee
• το γόνατό της = her knee

Literally, Greek uses:

• the knee her

This is normal Greek structure. Similar examples:

• το χέρι μου = my hand
• το πόδι σου = your leg
• το κεφάλι του = his head
• το χέρι τους = their hand/arm

So της is not to her here; it is a possessive clitic meaning her.

The basic word is indeed γόνατο.

When a word like γόνατο is followed by an enclitic such as μου, σου, του, της, μας, σας, τους, Greek often adds an extra written accent to help preserve the normal stress pattern.

So:

• γόνατο = basic form
• το γόνατό της = her knee

This happens because της is an enclitic. You will see the same thing in phrases like:

• το πρόσωπό μου = my face
• το τηλέφωνό σου = your phone

So the extra accent does not mean the word has changed meaning; it is mainly a stress/orthography issue.

Greek normally uses the definite article with possessed nouns.

So where English says:

• her knee

Greek usually says:

• το γόνατό της
• literally: the knee her

This is the standard Greek way to express possession with body parts and many other nouns.

More examples:

• το σπίτι μου = my house
• η μητέρα του = his mother
• τα βιβλία μας = our books

So the article is not optional here in normal usage.

Because γόνατο is simply a neuter noun in Greek. Noun gender in Greek is grammatical, so it does not always match anything logical or biological.

Its basic forms are:

• το γόνατο = the knee
• plural: τα γόνατα = the knees

Because it is neuter singular, words that agree with it also appear in neuter singular, which is why you get:

• πρησμένο (not πρησμένος or πρησμένη)

Πρησμένο is an adjective meaning swollen, and it agrees with γόνατο.

Since γόνατο is:

• neuter
• singular

the adjective must also be:

• neuter singular → πρησμένο

Compare:

• ο αστράγαλος είναι πρησμένος = the ankle is swollen
• η παλάμη είναι πρησμένη = the palm is swollen
• το γόνατο είναι πρησμένο = the knee is swollen

So this is standard adjective agreement.

Here λίγο means a little / slightly and works like an adverb, modifying πρησμένο.

So:

• λίγο πρησμένο = a little swollen / slightly swollen

It is not really describing the noun directly as a little knee or anything like that. It is qualifying the state.

Compare:

• είναι λίγο κουρασμένος = he is a little tired
• είναι λίγο δύσκολο = it is a little difficult

So in this sentence, λίγο softens the description.

The subject is το γόνατό της.

So the sentence structure is:

• το γόνατό της = her knee
• είναι λίγο πρησμένο = is a little swollen
• και χρειάζεται ξεκούραση = and needs rest

In other words:

• Her knee is a little swollen and needs rest.

Even though English speakers may first think about she, Greek is talking about the knee as the subject.

That is why both verbs are 3rd person singular:

• είναι
• χρειάζεται

The dictionary form is usually χρειάζομαι, which means to need.

This verb looks middle/passive in form, but in Modern Greek it often has an active meaning:

• χρειάζομαι = I need
• χρειάζεσαι = you need
• χρειάζεται = he/she/it needs

So here:

• το γόνατό της χρειάζεται ξεκούραση = her knee needs rest

You may also see the verb used with να + verb:

• Χρειάζεται να ξεκουραστεί. = She/It needs to rest.

But in this sentence it is followed by a noun:

• χρειάζεται ξεκούραση = needs rest

Because ξεκούραση here is being used in a general, uncountable sense: rest.

So:

• χρειάζεται ξεκούραση = needs rest

This is like English, where we also usually say needs rest, not needs a rest in this kind of medical/general statement.

Greek often omits the article with abstract or mass nouns when speaking generally:

• θέλω νερό = I want water
• χρειάζεται βοήθεια = he/she/it needs help
• χρειάζεται ξεκούραση = he/she/it needs rest

If you add an article, the meaning can become more specific depending on context.

By itself, χρειάζεται can mean:

• he needs
• she needs
• it needs

Greek often omits subject pronouns, so the exact subject comes from context.

In this sentence, the subject is clearly το γόνατό της (her knee), so here it means:

• it needs

If the subject were a woman, then χρειάζεται could mean she needs:

• Η Μαρία χρειάζεται ξεκούραση. = Maria needs rest.

Not completely. Greek word order is more flexible than English, although some versions sound more neutral than others.

The given sentence is a very natural, neutral way to say it:

• Μετά την πτώση, το γόνατό της είναι λίγο πρησμένο και χρειάζεται ξεκούραση.

You could move things around for emphasis, for example:

• Το γόνατό της είναι λίγο πρησμένο μετά την πτώση και χρειάζεται ξεκούραση.

But the original order is smoother because Μετά την πτώση sets the scene first.

What is less flexible is the relationship inside το γόνατό της. The possessive clitic της normally comes after the noun:

• το γόνατό της not
• της γόνατο

The phrase Μετά την πτώση is an introductory time phrase: After the fall.

Greek often puts a comma after such an opening phrase, just as English often does:

• After the fall, her knee is a little swollen...
• Μετά την πτώση, το γόνατό της...

The comma helps readability and marks a small pause. In some short sentences, punctuation can vary a bit, but here the comma is very natural.

Yes. Greek has several natural ways to express this idea.

The sentence you have is very straightforward and descriptive:

• είναι λίγο πρησμένο = is a little swollen

But Greek could also use other structures depending on style and context, for example:

• Το γόνατό της έχει πρηστεί λίγο. = Her knee has swollen a little.
• Το γόνατό της είναι πρησμένο. = Her knee is swollen.

The version in your sentence is especially clear for describing a present condition:

• it is a little swollen
• and it needs rest

So it sounds very natural in medical or everyday speech.