Við eigum eftir að fylla út eyðublaðið áður en fundurinn byrjar.

In this sentence, eiga eftir að + verb means something like to still have left to do or to still need to do.

So:

• Við eigum eftir að fylla út eyðublaðið
  = We still have to fill out the form / We have yet to fill out the form

This is a very common Icelandic structure. It does not literally mean possession here, even though eiga often means to own or to have.

A few similar examples:

• Ég á eftir að hringja.
  = I still have to call.
• Þau eiga eftir að klára verkefnið.
  = They still have to finish the project.

Because við is we, and the verb has to agree with the subject.

The infinitive is eiga, but here it is conjugated in the present tense:

• ég á = I have
• þú átt = you have
• hann/hún/það á = he/she/it has
• við eigum = we have
• þið eigið = you (plural) have
• þeir/þær/þau eiga = they have

So:

• Við eigum eftir að...
  = We still have to...

The að here is the infinitive marker, like to in English.

So:

• að fylla út = to fill out

After the expression eiga eftir, Icelandic normally uses að + infinitive:

• á eftir að fara = still has to go
• eigum eftir að lesa = still have to read

So the pattern is:

• subject + eiga + eftir + að + infinitive

Yes. Fylla út is a verb phrase meaning to fill out.

• fylla = fill
• út = out

Very often in Icelandic, a verb and a particle work together like this, similar to English phrasal verbs.

Here:

• fylla út eyðublaðið = fill out the form

You may also notice that the particle can sometimes appear in slightly different positions depending on the structure, but in this sentence fylla út is the normal form.

Eyðublaðið is the definite singular neuter form of eyðublað (form).

Breakdown:

• eyðublað = a form
• eyðublaðið = the form

The ending -ið is the definite article attached to the noun. Icelandic usually adds the to the end of the noun instead of using a separate word.

In this sentence, eyðublaðið is the object of fylla út, so it is in the accusative. For this noun, nominative and accusative singular happen to look the same:

• nominative: eyðublaðið
• accusative: eyðublaðið

So you do not see a visible change here, but grammatically it is still the object.

Icelandic often uses the present tense when English would use a future meaning after time words like before, when, after, and so on.

So:

• áður en fundurinn byrjar
  literally: before the meeting starts natural English: before the meeting starts/begins

Even though it refers to the future, Icelandic normally keeps the verb in the present tense here:

• þegar hann kemur = when he comes
• áður en við förum = before we go
• eftir að hún vaknar = after she wakes up

That is completely normal Icelandic grammar.

Fundurinn means the meeting.

Breakdown:

• fundur = meeting
• fundurinn = the meeting

It is the subject of byrjar:

• fundurinn byrjar = the meeting starts

So the whole clause:

• áður en fundurinn byrjar
  = before the meeting starts

This is just a subordinate clause introduced by áður en = before.

This is normal main-clause Icelandic word order.

In a regular statement, Icelandic usually puts the finite verb in second position:

• Við = subject
• eigum = finite verb
• eftir að fylla út eyðublaðið = the rest of the predicate

So the structure is very natural:

• Við | eigum | eftir að fylla út eyðublaðið

If something else came first for emphasis, the finite verb would still usually stay in second position. For example:

• Eyðublaðið eigum við eftir að fylla út.
  = The form, we still have to fill out.

That second-position rule is a major feature of Icelandic.

Not exactly. The emphasis is not just on future time, but on something that remains undone.

Við eigum eftir að fylla út eyðublaðið means:

• We still have to fill out the form
• We have yet to fill out the form

It suggests the task is not done yet.

If you simply wanted to say we will fill out the form later, you might choose a different wording depending on context, such as one with síðar (later) or a plain future meaning from the present tense.

So eiga eftir að specifically highlights remaining obligation / incompletion.

It can be followed by other tenses when the context requires them, but in sentences about the future, the present tense is very common and normal.

In your sentence:

• áður en fundurinn byrjar
  = before the meeting starts

That is the standard way to say it.

But with past reference, you can of course use past tense:

• Við fylltum út eyðublaðið áður en fundurinn byrjaði.
  = We filled out the form before the meeting started.

So the tense depends on the time reference of the sentence as a whole.

You would change við eigum to ég á:

• Ég á eftir að fylla út eyðublaðið áður en fundurinn byrjar.
  = I still have to fill out the form before the meeting starts.

That is a very useful pattern to practice:

• Ég á eftir að... = I still have to...
• Þú átt eftir að... = You still have to...
• Við eigum eftir að... = We still have to...

It is quite neutral and very natural Icelandic. It would work well in everyday speech, workplace communication, and ordinary writing.

Nothing in it is especially literary or especially slangy.

In fact, both of these parts are very common in normal Icelandic:

• eiga eftir að for still have to / have yet to
• áður en... byrjar for before ... starts

So this is a very useful real-life sentence pattern to learn.