Í hverfinu er rólegt á sunnudag.

Hverfi means neighborhood.
Hverfinu is the definite form in the dative singular: roughly (in) the neighborhood.

• hverfi = a neighborhood (indefinite)
• hverfið = the neighborhood (nominative/accusative)
• hverfinu = (in/from) the neighborhood (dative)

Here, the ending -inu shows dative + definite.

The preposition í takes different cases depending on meaning:

• í + dative = location/state (in, no movement)
• í + accusative = motion into (into, movement)

So:

• Í hverfinu = in the neighborhood (location)
• Í hverfið would imply movement: into the neighborhood

Yes. Icelandic is generally V2 (verb-second) in main clauses: the finite verb (here er) tends to come second, no matter what comes first.

So if you front the location Í hverfinu, the verb follows:

• Í hverfinu | er | rólegt ...

You could also start differently, e.g.:

• Það er rólegt í hverfinu á sunnudag.
• Á sunnudag er rólegt í hverfinu.

All are grammatical; they just highlight different parts.

This is an impersonal construction. Icelandic often says “it is X” without a concrete subject. You can think of an implied dummy it.

A very common alternative is to include the dummy pronoun explicitly:

• Það er rólegt í hverfinu á sunnudag. = It is quiet ...

In your sentence, það is simply left out.

Because rólegt is not describing hverfið as a noun phrase directly here; it’s used in an impersonal “it is quiet” pattern.

In that pattern, the adjective is typically:

• neuter singular: rólegt, kalt, gott, etc.

So:

• (Það) er rólegt. = It is quiet.

If you instead make the neighborhood the grammatical subject, you’d get agreement with hverfið:

• Hverfið er rólegt á sunnudag.
  Here hverfið (neuter singular) still leads to rólegt, but for a different reason: normal subject–predicate agreement.

It’s an adjective used as a predicate adjective (describing the situation/state).

Compare:

• Það er rólegt. (adjective: “It is quiet.”)
• Hann talar rólega. (adverb: “He speaks quietly.”)

So rólega would be the adverb form, but that would change the structure and meaning.

For days/dates as time expressions, Icelandic commonly uses á + accusative to mean on (a given day):

• á mánudag, á þriðjudag, á sunnudag

So sunnudag is the accusative singular of sunnudagur.

Often, á sunnudag is understood as on Sunday (a specific upcoming/particular Sunday) depending on context.

If you clearly mean every Sunday / on Sundays (habitually), Icelandic very often uses the plural:

• á sunnudögum = on Sundays

So:

• Í hverfinu er rólegt á sunnudögum. = “It’s quiet in the neighborhood on Sundays.”

In Icelandic, days of the week (and months) are not capitalized unless they start a sentence.

So:

• á sunnudag (normal)
• Sunnudag only if it’s the first word in the sentence.

Negation usually places ekki after the verb (and often after the subject if there is one):

• Í hverfinu er ekki rólegt á sunnudag. = “It isn’t quiet ...”

Yes/no question: put the verb first:

• Er rólegt í hverfinu á sunnudag? = “Is it quiet ...?”

A few common ones:

• Í: like a long ee sound (roughly /iː/).
• hv in hverfinu: pronounced like kv for many speakers (so kver-).
• rólegt: stress on the first syllable (RÓ-legt); the ó is a long vowel.
• sunnudag: also first-syllable stress (SUNN-u-dag); final -g is often fairly soft in casual speech.