Við höfum beðið lengi í dag.

• Við = we
• höfum = have (the 1st person plural present form of hafa, to have)
• beðið = the form of bíða (to wait) used after hafa
• lengi = for a long time / long
• í dag = today

So the sentence is literally something like We have waited long today, which in natural English is We have waited a long time today or We have been waiting a long time today.

This is the Icelandic perfect tense, formed with:

• hafa in the present tense
• plus the main verb in the supine form

So:

• við höfum = we have
• beðið = waited

That is the normal way to say have waited / have been waiting in Icelandic.

Icelandic does not have a separate continuous form like English have been waiting in the same way English does.

Because of that, Við höfum beðið lengi í dag can cover both ideas:

• We have waited a long time today
• We have been waiting a long time today

In context, English will often prefer have been waiting, but Icelandic usually just uses the perfect form with hafa + supine.

Because after hafa in the perfect tense, Icelandic uses the verb’s supine form.

Compare:

• bíða = to wait (infinitive)
• bíðum = we wait / are waiting (present tense)
• beðið = the form used after hafa in the perfect

So:

• Við bíðum = We are waiting / We wait
• Við höfum beðið = We have waited / have been waiting

You may also notice that the vowel changes from í to ei: bíða → beðið. That is just part of how this verb forms its perfect.

Lengi is an adverb meaning for a long time or long in a time-duration sense.

So Icelandic often says:

• Við höfum beðið lengi
  = We have waited for a long time

English uses for, but Icelandic does not need a separate preposition here. The idea of duration is already built into lengi.

Because today is usually seen as a time period that is still continuing.

If the day is not over yet, the perfect is very natural:

• Við höfum beðið lengi í dag
  = We have waited / have been waiting a long time today

This is similar to English, where today often works well with the present perfect because it still includes now.

If the time period were clearly finished, Icelandic would often use the past tense instead, for example:

• Við biðum lengi í gær
  = We waited a long time yesterday

Yes. Icelandic word order is somewhat flexible, but in a main clause the finite verb usually stays in second position.

The version you have is the neutral order:

• Við höfum beðið lengi í dag

You can also move í dag to the front:

• Í dag höfum við beðið lengi

That is still correct, but once í dag comes first, höfum must come next. This is a very common Icelandic pattern.

Yes. When you are waiting for someone or something, bíða often takes a genitive object.

A useful pattern is:

• að bíða einhvers = to wait for someone/something

In your sentence, no object is stated, so it simply means we have waited.

But you could say, for example:

• Við höfum beðið hennar lengi í dag
  = We have waited for her a long time today

This is worth remembering because the case after bíða is not what an English speaker would automatically expect.