Ég verð fegin þegar ég er komin heim og þessi vika er búin.

Because fegin is the feminine singular form.

The sentence tells you that the speaker is female:

• Ég verð fegin = I will be glad / relieved, said by a woman
• A man would say Ég verð feginn

This kind of agreement is very common in Icelandic. Adjectives and participle-like forms often agree with the person or noun they describe in gender, number, and case.

Verð is the 1st person singular present of verða, which often means become.

So:

• Ég verð fegin literally means I become glad
• In natural English, that is usually I’ll be glad or I’ll be relieved

Even though the verb is formally present tense, this sentence has a future meaning because it is talking about how the speaker will feel once certain things have happened.

Yes, þegar can mean two different things:

• þegar = when as a conjunction
• þegar = already as an adverb

In this sentence, it is clearly the conjunction when, because it introduces a clause:

• þegar ég er komin heim ... = when I have gotten home ...

So here þegar is not already.

This is very normal in Icelandic.

After time words like þegar = when, Icelandic often uses present tense forms to talk about the future, especially when English would say something like:

• when I get home
• when the week is over

So:

• þegar ég er komin heim = when I have gotten home / when I’m home
• þegar þessi vika er búin = when this week is over

The main clause carries the future feeling:

• Ég verð fegin ... = I’ll be glad ...

This is similar to English, which also says when I get home, not when I will get home.

It is a very common Icelandic way to express having arrived somewhere, especially with verbs of motion.

• koma = come
• ég er komin heim literally looks like I am come home
• natural English: I’ve gotten home, I’ve arrived home, or simply I’m home now

The important idea is completed movement with a resulting state:

• the speaker has arrived
• and is now at home

The form komin agrees with the female speaker. A male speaker would say kominn.

Because Icelandic makes an important distinction:

• heim = homeward / to home, showing movement toward home
• heima = at home, showing location

Since the sentence is about getting home, Icelandic uses heim:

• ég er komin heim = I have gotten home

If you were just describing location, you would use heima:

• Ég er heima = I am at home

This is a very important pair to learn.

Here búin means finished, over, or done.

So:

• þessi vika er búin = this week is over / this week is finished

The word agrees with vika, which is a feminine singular noun, so the form is búin.

Compare:

• dagurinn er búinn = the day is over
• árið er búið = the year is over
• vikan er búin = the week is over

So the form changes depending on the noun it describes.

It is closely related, but the construction is different.

You may have seen:

• Ég er búin að lesa bókina = I have finished reading the book

That pattern is:

• vera búinn/búin/búið að + infinitive

But in your sentence, there is no infinitive after it:

• þessi vika er búin

Here it simply means is over / is finished.

So the same word family is involved, but the grammar is not exactly the same.

Because þessi vika is the subject of the clause:

• þessi vika er búin = this week is over

Subjects are normally in the nominative case, so you get:

• þessi = nominative feminine singular
• vika = nominative feminine singular

If it were an object, you might see a different case, such as þessa viku.

Yes.

The structure is:

• Ég verð fegin = main clause
• þegar ég er komin heim og þessi vika er búin = time clause

So the meaning is:

• I’ll be glad when both of these are true:
  1. I have gotten home
  2. this week is over

You can think of it as shorthand for:

• þegar ég er komin heim og þegar þessi vika er búin

The second þegar is simply left out because it is understood.

Only the parts that refer to the speaker would change.

A man would say:

Ég verð feginn þegar ég er kominn heim og þessi vika er búin.

Changes:

• fegin → feginn
• komin → kominn

But búin does not change, because it describes vika, and vika is still feminine.

So the agreement works like this:

• words about I change with the speaker’s gender
• words about week change with the noun vika