Við tölum lágt í bókasafninu.

Við means we.

Yes, in a normal sentence like this, you usually include it. Even though the verb form tölum already shows we, Icelandic normally keeps the subject pronoun in ordinary statements.

So:

• Við tölum lágt = We speak quietly
• not usually just Tölum lágt unless the context is special, poetic, or elliptical

Tölum is the 1st person plural present form of the verb tala, which means to speak or to talk.

The pattern is:

• ég tala = I speak
• þú talar = you speak
• við tölum = we speak

So the ending -um here tells you the subject is we.

Also notice that the stem vowel changes slightly: tala → tölum. This kind of vowel change is common in Icelandic verb forms.

Lágt comes from the adjective lágur, which basically means low. In this sentence, it is used adverbially, so it means in a low voice, softly, or quietly.

So Við tölum lágt is literally something like We speak low, but in natural English that becomes We speak quietly or We speak softly.

This is very natural in Icelandic.

Here it functions as an adverb because it describes how we speak.

A very common Icelandic pattern is that the neuter singular form of an adjective is used adverbially.

So:

• lágur = low
• lágt = low / quietly / softly, when used adverbially

That is why lágt appears in this form.

Because í can take different cases depending on meaning.

Here, í means in, showing a location: we are speaking in the library. With location, í takes the dative case.

So:

• í bókasafninu = in the library
  dative, because it is a location

But if you were talking about motion into the library, you would use the accusative:

• í bókasafnið = into the library

This location-vs-motion difference is very important in Icelandic.

It shows both:

1. the definite article = the
2. the dative singular ending

The base noun is:

• bókasafn = library

This is a neuter noun. Its definite form is built onto the end of the word, not written as a separate word like English the.

So:

• bókasafn = library
• bókasafnið = the library
• bókasafninu = the library, in the dative case

That is why Icelandic learners often say the article is attached to the noun.

You would say í bókasafni.

Compare:

• í bókasafni = in a library
• í bókasafninu = in the library

Both are dative after í for location, but the second one has the definite ending meaning the.

Yes. The sentence as given has a very normal word order:

• Við tölum lágt í bókasafninu.

But Icelandic allows other orders for emphasis or style. For example:

• Í bókasafninu tölum við lágt. = In the library, we speak quietly.

When something else is moved to the front, Icelandic usually keeps the finite verb in second position. This is often called the V2 rule.

So after fronting Í bókasafninu, the verb tölum comes next, and then við follows.

Usually it can mean either, depending on context.

• Við tölum lágt í bókasafninu. can mean We speak quietly in the library as a general rule or habit.
• It can also describe what is happening now, if the situation makes that clear.

If you want to be more explicitly progressive, Icelandic often uses vera að + infinitive:

• Við erum að tala lágt í bókasafninu. = We are speaking quietly in the library.

So the simple present is flexible, and context matters.

Yes. Bókasafn is a compound noun:

• bók = book
• safn = collection

So bókasafn is literally something like book collection, which is the Icelandic word for library.

Compounds are extremely common in Icelandic, so noticing the parts of a word can often help you remember its meaning.