Það þarf að vekja hana snemma á virkum dögum, því hún geispar allan morguninn ef hún sefur of lengi.

This is an impersonal construction.

• það is a dummy subject, like English it in it is necessary to...
• þarf is the 3rd person singular present of þurfa (to need / to have to)

So Það þarf að vekja hana literally means It is necessary to wake her.
In more natural English, you might translate it as She needs to be woken.

After þarf in this kind of construction, Icelandic normally uses að + infinitive.

• vekja = to wake (someone)
• að vekja = to wake

So:

• það þarf að vekja hana = it is necessary to wake her

This is a very common and important distinction:

• vekja = to wake someone up
  
  • transitive: it takes an object
• vakna = to wake up
  
  • intransitive: no direct object

So:

• Ég vek hana. = I wake her.
• Hún vaknar. = She wakes up.

In your sentence, vekja is used because someone is waking her.

Because hana is the accusative form of the pronoun hún.

The verb vekja takes a direct object, and direct objects are often in the accusative.

Pronoun forms:

• hún = she (nominative)
• hana = her (accusative)
• henni = her (dative)
• hennar = her / hers (genitive)

So:

• hún wakes someone
• someone wakes hana

Snemma is an adverb, meaning early.

It describes when she is woken:

• vekja hana snemma = wake her early

Because it is an adverb, it does not change for gender, number, or case.

Here á means on, in the sense of time: on weekdays.

• dögum is the dative plural of dagur (day)
• virkum agrees with it, so it is also dative plural

So:

• virkir dagar = weekdays / working days
• á virkum dögum = on weekdays

This is a fixed and very common time expression.

Yes — því can mean different things depending on context.

In this sentence, því is a conjunction meaning:

• because
• sometimes for

So here:

• ..., því hún geispar... = ..., because she yawns...

This is different from other uses of því as a pronoun form.

Because því and ef introduce subordinate clauses.

In main clauses, Icelandic often has verb-second word order.
But in subordinate clauses, the word order is usually more like:

• conjunction + subject + verb

So:

• því hún geispar = because she yawns
• ef hún sefur of lengi = if she sleeps too long

Not:

• því geispar hún
• ef sefur hún

This is a very common pattern in Icelandic.

Geispar is the 3rd person singular present form of geispa (to yawn).

So:

• ég geispa = I yawn
• hún geispar = she yawns

In this sentence:

• hún geispar allan morguninn = she yawns all morning

This is an example of the accusative of time duration.

Icelandic often uses the accusative to show how long something lasts.

So:

• allan morguninn = all morning

Both words are masculine accusative singular:

• allan = accusative form of allur (all)
• morguninn = the morning in accusative

So the phrase literally means something like the whole morning.

Because Icelandic uses the present tense for general or repeated conditions, just like English does.

So:

• ef hún sefur of lengi = if she sleeps too long

This does not mean only right now. It can describe a general pattern:

• whenever she sleeps too long, she yawns all morning

Also, sefur is the 3rd person singular present of sofa (to sleep).

Of means too, and lengi means for a long time / long in an adverbial sense.

So:

• of lengi = too long

Lengi is used because it describes the duration of sleeping, so Icelandic needs an adverb, not an adjective.

Compare:

• langur dagur = a long day
  
  • adjective describing a noun
• sofa lengi = sleep for a long time
  
  • adverb describing the verb

So hún sefur of lengi means she sleeps too long.

Yes. That is often the most natural English translation.

Even though Icelandic uses the active-looking impersonal construction:

• Það þarf að vekja hana snemma

English often prefers:

• She needs to be woken early

So the Icelandic structure is not literally passive, but it often corresponds to a passive-style translation in English.