Eftir langan vinnudag vil ég bara fara heim og sofa.

Because Icelandic is a verb-second language in main clauses.

In this sentence, Eftir langan vinnudag comes first. Once that first position is taken, the finite verb vil has to come next, and the subject ég follows it.

So the structure is:

• Eftir langan vinnudag = first element
• vil = finite verb in second position
• ég = subject

If you started the sentence with the subject, then ég vil would be correct:

• Ég vil bara fara heim og sofa.

The basic forms are:

• langur = long
• vinnudagur = workday

In this sentence, that phrase appears in the accusative singular masculine, so both words change form:

• langur → langan
• vinnudagur → vinnudag

This is a very important Icelandic pattern: adjectives must agree with the noun in gender, number, and case.

Vinnudagur is a compound noun meaning workday or working day.

It is made from:

• vinna = work
• dagur = day

Icelandic uses compound words very freely, so this kind of formation is extremely common.

After vilja = to want, Icelandic normally uses a bare infinitive, without að.

So you get:

• ég vil fara = I want to go
• ég vil sofa = I want to sleep

That is why the sentence has:

• vil ég bara fara heim og sofa

not a form with að before those verbs.

Both fara and sofa depend on the finite verb vil.

So the structure is basically:

• vil = want
• fara heim = go home
• og sofa = and sleep

In other words, Icelandic is using one finite verb plus two coordinated infinitives.

Because heim and heima are not the same.

• heim = home, toward home
• heima = at home

So:

• Ég fer heim. = I go home.
• Ég er heima. = I am at home.

Since fara expresses movement, heim is the correct form here.

Bara usually means just, only, or simply.

Here it adds the idea that the speaker wants nothing more complicated than this: just go home and sleep.

Its position is very natural. In Icelandic, bara is quite flexible, but moving it can slightly change what gets emphasized.

Yes. In this sentence, eftir is a preposition meaning after in a time sense.

It sets the time background for the rest of the sentence:

• Eftir langan vinnudag = after a long workday

So the sentence first gives the situation, and then says what the speaker wants.

In normal modern Icelandic, the subject pronoun is usually not omitted.

So ég is normally kept:

• vil ég = I want

Even though Icelandic verbs do carry grammatical information, Icelandic is not a language where subject pronouns are regularly dropped in ordinary speech.

Yes. There are several natural alternatives, depending on style and emphasis. For example:

• Ég vil bara fara heim og sofa eftir langan vinnudag.
• Eftir langan dag í vinnunni vil ég bara fara heim og sofa.
• Eftir vinnuna vil ég bara fara heim og sofa.

The original sentence is natural and idiomatic, but Icelandic gives you more than one good way to express the same basic idea.