Dagdvölin er opin í dag.

Because opin agrees with Dagdvölin, which is a feminine singular noun.

In Icelandic, adjectives usually change form to match the noun’s:

• gender
• number
• case

Here the pattern is:

• opinn = masculine singular
• opin = feminine singular
• opið = neuter singular

So if the subject is Dagdvölin and that noun is feminine, the adjective must be opin.

Er means is. It is the present tense, 3rd person singular form of the verb að vera (to be).

So:

• ég er = I am
• þú ert = you are
• hann/hún/það er = he/she/it is

In this sentence, er links the subject Dagdvölin with the description opin.

Because Icelandic usually does not need a separate dummy subject like English it in sentences such as It is open when a specific noun is already the subject.

Here, the subject is explicitly named:

• Dagdvölin er opin

So Icelandic says literally:

• Dagdvölin is open

There is no need to add an extra pronoun.

Í dag literally means in day, but idiomatically it means today.

This is just the normal Icelandic expression:

• í dag = today
• í morgun = this morning / in the morning

The preposition í often means in, but in fixed time expressions it may not translate word-for-word into natural English.

Because dag is part of the fixed expression í dag, and that expression normally appears without the definite article.

By contrast, Dagdvölin has the definite article attached at the end:

• dagdvöl = day care / day stay / day program
• dagdvölin = the day care / the day program

This is very common in Icelandic: the definite article is often attached to the noun as a suffix.

It could be understood either way depending on context, but grammatically it looks like an ordinary noun with the suffixed definite article:

• dagdvöl = day care, daytime stay, day program
• dagdvölin = the day care / the daytime stay / the day program

In real-life usage, something like Dagdvölin could also function as the name of a specific institution or service. Icelandic often capitalizes the first word of a sentence anyway, so capitalization alone does not prove it is a proper name.

This is the most neutral word order:

• subject + verb + complement + time expression

So:

• Dagdvölin = subject
• er = verb
• opin = subject complement
• í dag = time expression

Icelandic word order is flexible, and you could move í dag for emphasis:

• Í dag er Dagdvölin opin.

That version puts more focus on today.

It is in the nominative singular, because it is the subject of the sentence.

The basic form is:

• dagdvöl = nominative singular, indefinite
• dagdvölin = nominative singular, definite

Since the sentence says that Dagdvölin is open, it is functioning as the subject, so nominative is the expected case.

Here dag is in the accusative singular.

The noun dagur (day) has forms such as:

• nominative: dagur
• accusative: dag
• dative: degi
• genitive: dags

The expression í dag uses the accusative. Many Icelandic prepositions govern particular cases, and í can take either accusative or dative depending on meaning. In this time expression, you simply learn the phrase as í dag.

No, not in normal Icelandic for this meaning.

To say something is open, Icelandic simply uses:

• er opin

You do not normally add að vera after er here. That would sound unnatural.

So the natural sentence is:

• Dagdvölin er opin í dag.

The adjective would change to match the noun’s gender.

Examples:

• masculine: Skólinn er opinn í dag. = The school is open today.
• feminine: Verslunin er opin í dag. = The shop is open today.
• neuter: Safnið er opið í dag. = The museum is open today.

This is a very important Icelandic pattern:

• opinn with masculine
• opin with feminine
• opið with neuter

A rough guide for an English speaker is:

• Dagd- starts like dahgth or dahkth depending on speech rhythm
• -vö- has a vowel that does not exist exactly in English; it is like German ö
• -lin sounds roughly like lin

A very rough approximation might be:

• DAHG-thvuh-lin

But that is only approximate. A few useful points:

• a in dag is like the a in British father
• ö is a front rounded vowel, similar to German ö
• dv can be a bit tricky for English speakers and may sound more blended in fast speech

If you want good pronunciation, it is best to listen to native audio, because Icelandic vowels and consonant combinations are often not obvious from spelling alone.

Yes, usually í dag suggests a time-specific situation: it is open today, perhaps whether or not that is always true.

Without í dag, the sentence:

• Dagdvölin er opin.

would simply mean:

• The day care is open.

Adding í dag gives it a more specific time frame:

• The day care is open today.

Yes. You can say:

• Í dag er Dagdvölin opin.

This is perfectly natural and puts more emphasis on today.

When a phrase like í dag comes first, the verb still stays in the second position, which is a common Icelandic word-order rule:

• Í dag
  • er
    • Dagdvölin
      • opin

So that version is a good example of Icelandic verb-second word order.