Í maí og júní dreymir mig oft um ferðalag, jafnvel þegar ég hef mikið að gera.

Because dreyma is not used like the English verb to dream.

In this common Icelandic pattern, the person who has the dream is in the accusative, not the nominative. So Icelandic says:

• mig dreymir = I dream / I am dreaming
• literally, something closer to me dreams

So mig is correct here, not ég.

This is because of Icelandic verb-second word order.

The sentence begins with the time expression Í maí og júní. When something other than the subject comes first, the finite verb usually comes next:

• Mig dreymir oft um ferðalag.
• Í maí og júní dreymir mig oft um ferðalag.

So dreymir moves into the second position, and mig comes after it.

This means in May and June.

The preposition í is used for time expressions like months, meaning in. With months, it normally goes with the dative, although with maí and júní the form looks the same as the dictionary form, so you do not see a visible change here.

So:

• í maí = in May
• í júní = in June

Because in Icelandic, names of months are normally written with lower-case letters, unlike in English.

So Icelandic writes:

• maí
• júní

not Maí or Júní, unless the word begins a sentence.

Oft means often, and it is an adverb of frequency.

In Icelandic, adverbs like this commonly come after the finite verb and before longer phrases such as um ferðalag. So dreymir mig oft um ferðalag is very natural.

You may see some flexibility in adverb placement, but this position is standard and idiomatic.

Because the verb is commonly used with um when you mean dream about something:

• dreyma um eitthvað = to dream about something

The preposition um takes the accusative, so ferðalag is in the accusative here. Since ferðalag is a neuter noun, its accusative singular looks the same as its nominative singular.

Because the sentence is talking about a trip / travel in a general, indefinite way, not about one specific trip already known to the listener.

So:

• um ferðalag = about a trip / about travel
• um ferðalagið would mean about the trip

The version without the article sounds more general and natural here.

Ferðalag means something like journey, trip, or travel.

In this sentence it suggests the idea of going away or taking a trip, rather than focusing on one tiny act of movement. It has a slightly broader feel than just ferð in many contexts.

So um ferðalag here is a natural way to express dreaming about going on a trip.

Jafnvel means even, and þegar means when.

Together, jafnvel þegar means even when.

It adds a contrast: the speaker dreams about travel not only in easy or relaxed moments, but even when they are busy.

This is a very common Icelandic expression meaning I have a lot to do or I am busy.

Literally:

• ég hef = I have
• mikið = much / a lot
• að gera = to do

So the whole phrase means that the speaker has many things to do.

Also, mikið is in the neuter singular because it is being used as a general quantity word, not as an adjective agreeing with a specific noun.

Yes, it is normal here.

Icelandic often uses a comma before a subordinate clause, especially one introduced by a word like þegar. So the comma helps mark the shift from the main clause:

• Í maí og júní dreymir mig oft um ferðalag

to the subordinate clause:

• jafnvel þegar ég hef mikið að gera

So the punctuation is natural and standard.

You could, but it would likely suggest a different meaning or create ambiguity.

• Í maí og júní dreymir mig oft um ferðalag clearly means during May and June, I often dream about travel.
• Mig dreymir oft um ferðalag í maí og júní could sound more like the trip itself is in May and June.

So the original word order is better if you want the time phrase to describe when the dreaming happens.