Learn Hardcore Icelandic: Hann er hljóðlátur í bókasafninu.

In the sentence Hann er hljóðlátur í bókasafninu, the subject is Hann (he), which is masculine and in the nominative case, so the adjective also has to be masculine nominative: hljóðlátur.

If the person was a woman, would the sentence change?

Yes, the pronoun and the adjective would both change:

Hann er hljóðlátur í bókasafninu. – He is quiet in the library.
Hún er hljóðlát í bókasafninu. – She is quiet in the library.

Here Hún is feminine, so the adjective must also be feminine: hljóðlát (no -ur). The rest of the sentence stays the same.

How would I say They are quiet in the library?

You need plural pronouns and plural adjective forms:

Masculine group (all or mostly men):
Þeir eru hljóðlátir í bókasafninu.
Feminine group (all women):
Þær eru hljóðlátar í bókasafninu.
Neuter / mixed or unspecified group:
Þau eru hljóðlát í bókasafninu.

Notice how hljóðlátur changes:

hljóðlátir (masc. plural)
hljóðlátar (fem. plural)
hljóðlát (neuter plural – same form as fem. singular).

Why is it í bókasafninu and not á bókasafninu?

Both í and á can mean something like in / at, but they’re used with different kinds of places:

í is more literally in, inside something.
á is more like on, at, commonly used with open places, surfaces, events, etc.

A bókasafn (library) is a building/room you go inside, so you normally say:

í bókasafninu – in the library

You might use á with places like á kaffihúsinu (at the café), á tónleikum (at a concert), etc.

Why does bókasafninu have that ending -inu? What happened to bókasafn?

Several things are going on in bókasafninu:

bókasafn – base noun, a library (neuter).
The preposition í meaning in a place of rest takes the dative case, so bókasafn becomes bókasafni in the dative singular.
The definite article (the) is attached as a suffix in Icelandic. For neuter dative singular, that suffix is -nu.

So:

bókasafn (nom. sg., a library)
bókasafni (dat. sg., in a library)
bókasafni + nu → bókasafninu (dat. sg. definite: in the library)

The small extra u is added for pronunciation and spelling reasons, but historically it’s bókasafni + nu.

Why is the case dative here? Does í always take the dative?

Í can take either dative or accusative, depending on the meaning:

Í + dative → location / state: in (where something is)
- Hann er í bókasafninu. – He is in the library.
Í + accusative → movement into something: into (where something is going)
- Hann fer í bókasafnið. – He goes into the library.

In Hann er hljóðlátur í bókasafninu, we are talking about where he is (not where he is going), so dative is used: bókasafninu.

What is the difference between hljóðlátur and rólegur? Both seem to mean quiet/calm.

They overlap, but the nuance is slightly different:

hljóðlátur – literally sound-quiet:
- focuses on someone not making (much) noise, speaking softly, not talking a lot.
rólegur – calm, relaxed:
- focuses more on being calm, not stressed or agitated.

In a library, hljóðlátur fits very well because it emphasizes not speaking loudly or making noise.
You could also say:

Hann er rólegur í bókasafninu. – He is calm in the library.

That sounds more like he feels relaxed there, rather than specifically being quiet out of consideration.

Is the word order fixed? Could I say Í bókasafninu er hann hljóðlátur?

Yes, Icelandic word order is fairly flexible, and your alternative is possible:

Hann er hljóðlátur í bókasafninu. – neutral, normal order.
Í bókasafninu er hann hljóðlátur. – puts extra emphasis on in the library.

Both obey the basic rule that the finite verb (er) usually stays in second position in main clauses. When you move Í bókasafninu to the front, er must still come second, so it becomes:

Í bókasafninu er hann hljóðlátur.

Can Icelandic drop the Hann, like some languages drop subject pronouns?

Normally, no. Icelandic is not a “pronoun-dropping” language like Spanish or Italian. The subject pronoun is usually required:

Hann er hljóðlátur í bókasafninu. – correct.
Er hljóðlátur í bókasafninu. – sounds incomplete / ungrammatical in normal speech.

You can omit pronouns sometimes in imperative constructions or very informal ellipsis, but for regular statements like this, keep Hann.

How do you pronounce hljóðlátur and bókasafninu?

Approximate guide (not strict IPA, just learner-friendly):

hljóðlátur
- hljóð- → like hlyo-th (with ð as the soft th in this).
- -lát- → like lout (but with a clearer a).
- -ur → reduced, like a short oor / uhr.
- Stress is on the first syllable: HLJÓÐ-lát-ur.
bókasafninu
- bó- → like bo in boat, with a long ó.
- -ka- → like ka (as in car but shorter).
- -safn- → roughly savtn (the f and n often blend).
- -inu → i-nu, with a short i like in bit.
- Stress again on the first syllable: BÓ-ka-saf-ni-nu.

In Icelandic, stress is almost always on the first syllable of a word.

How would I say this in the past tense: He was quiet in the library?

You only need to change the verb er (is) to its past form var (was):

Hann er hljóðlátur í bókasafninu. – He is quiet in the library.
Hann var hljóðlátur í bókasafninu. – He was quiet in the library.

The adjective hljóðlátur and the prepositional phrase í bókasafninu stay the same.

Can I add very to say He is very quiet in the library?

Yes, use mjög (very) before the adjective:

Hann er mjög hljóðlátur í bókasafninu. – He is very quiet in the library.

Typical adverb order is:

subject + verb + adverb + adjective + rest of the sentence
→ Hann er mjög hljóðlátur í bókasafninu.