Við kveðjum gestina við hurðina eftir að partíið er búið.

No. They just happen to look the same.

• Við at the beginning means we. It is the pronoun.
• við before hurðina is a preposition meaning by, at, near.

So:

• Við kveðjum... = We say goodbye...
• við hurðina = at/by the door

This is a very common thing in Icelandic: the same spelling can belong to different word classes.

Because kveðjum is the finite verb form that matches við = we.

The verb is að kveðja = to greet / to say goodbye to.
In the present tense:

• ég kveð
• þú kveður
• hann/hún/það kveður
• við kveðjum
• þið kveðið
• þeir/þær/þau kveðja

So við kveðjum means we say goodbye.

The form kveðja by itself is the infinitive, to say goodbye, not the form you use after we.

Because gestina is the accusative plural definite form, and here it is the direct object of kveðjum.

The noun is:

• gestur = guest
• plural nominative: gestir = guests
• plural accusative definite: gestina = the guests

In this sentence, the guests are the people being said goodbye to, so Icelandic uses the object form.

Compare:

• Gestirnir koma. = The guests are coming.
  Here they are the subject, so nominative is used.
• Við kveðjum gestina. = We say goodbye to the guests.
  Here they are the object, so accusative is used.

Hurðina is the accusative singular definite form of hurð = door.

It appears after the preposition við:

• við hurðina = at/by the door

The preposition við normally takes the accusative. So even though English just says at the door, Icelandic requires the accusative form:

• hurð = door
• hurðin = the door
• hurðina = the door (accusative)

So this is not because door is the object of the verb. It is because the preposition við requires that case.

Here it means say goodbye to.

The verb að kveðja can refer to greeting or taking leave depending on context, but in a sentence like this:

• Við kveðjum gestina...
• especially with eftir að partíið er búið

the natural meaning is we say goodbye to the guests.

So the context makes it clear that this is about people leaving after the party.

Eftir að means after when it introduces a whole clause.

Here the clause is:

• að partíið er búið = that the party is finished / over

So:

• eftir að partíið er búið = after the party is over

This is different from using eftir with just a noun phrase. For example:

• eftir partíið = after the party
• eftir að partíið er búið = after the party is finished/over

The second version is a full clause and feels a bit more explicit.

Because partíið means the party.

The base noun is:

• partí = party

It is a neuter noun, and the definite article is attached to the end of the noun in Icelandic:

• partí = a party
• partíið = the party

That final -ið is the attached definite article for this form.

Literally, er búið means is finished or is over.

• er = is
• búið = finished / completed (the neuter singular form of búinn)

So:

• partíið er búið = the party is over / finished

Icelandic often uses vera + búinn to express that something has been completed or is over.

Because partíið is a neuter singular noun, the adjective must match it:

• masculine: búinn
• feminine: búin
• neuter: búið

That is why it is er búið, not er búinn or er búin.

No, they only look similar.

• In partíið, -ið is the definite article: the party.
• In búið, the ending is part of the adjective form búinn/búin/búið, and here it is the neuter singular form.

So:

• partíið = noun + the
• búið = adjective agreeing with a neuter singular noun

They are different grammatical endings, even though they are spelled the same way here.

Because this is a subordinate clause introduced by að.

In Icelandic, subordinate clauses usually keep a more straightforward order:

• subject + verb + complement

So:

• partíið er búið
• subject: partíið
• verb: er
• complement: búið

That is the normal order here.

It can be understood as either by the door or at the door, depending on context. In natural English, at the door is often the best translation in this sentence.

So:

• Við kveðjum gestina við hurðina
  = We say goodbye to the guests at the door

The Icelandic phrase suggests the location where the goodbye happens, not movement toward the door.

Yes. The present tense in Icelandic can describe:

• something happening now
• a general habit
• a regular or typical action

So without more context, Við kveðjum gestina við hurðina eftir að partíið er búið can sound like:

• We say goodbye to the guests at the door after the party is over
• or We usually say goodbye to the guests at the door after the party is over

Context tells you whether it is a one-time present event or a habitual statement.

Because Icelandic handles that idea through the verb and case, not with a separate preposition like English does.

In English, you say:

• say goodbye to the guests

In Icelandic, the verb að kveðja directly takes an object:

• kveðja gestina

So the meaning of English to is built into how the Icelandic verb works with its object.