Við setjum nýja mottu í ganginn og hengjum úlpurnar á snagann.

They are the 1st person plural present tense forms of the verbs:

• setja = to put, place
• hengja = to hang

Because the subject is við = we, Icelandic uses:

• við setjum = we put / we are putting
• við hengjum = we hang / we are hanging

So the sentence is describing what we do.

Because the same subject applies to both verbs.

• Við setjum ... og hengjum ...

This works just like English:

• We put ... and hang ...

You could repeat við, but it is normally unnecessary when the subject stays the same.

Because mottu is in the accusative singular, and the adjective has to match it.

Here, mottu is the direct object of setjum:

• setja eitthvað = to put something

The noun is:

• motta = mat, rug

In this sentence it becomes:

• mottu = accusative singular

The adjective nýr / ný / nýtt = new must agree with the noun in gender, number, and case, so with feminine accusative singular it becomes:

• nýja mottu = a new mat

So the ending is there because of case agreement.

Because í can take either:

• dative for location: in
• accusative for motion into something: into

Here the mat is being placed into the hallway, so Icelandic uses the accusative:

• í ganginn = into the hallway

Compare:

• Mottan er í ganginum. = The mat is in the hallway.
  → location, so dative
• Við setjum mottu í ganginn. = We put a mat into the hallway.
  → motion, so accusative

For the same reason as í ganginn.

The preposition á can also take:

• dative for location: on
• accusative for motion onto something: onto

Since the coats are being hung onto the coat rack/peg, Icelandic uses the accusative:

• á snagann = onto the peg / coat rack

Compare:

• Úlpurnar eru á snaganum. = The coats are on the peg/rack.
  → location, dative
• Við hengjum úlpurnar á snagann. = We hang the coats on/onto the peg/rack.
  → motion, accusative

Úlpurnar means the coats.

The base noun is:

• úlpa = coat, parka

Plural:

• úlpur = coats

Definite plural:

• úlpurnar = the coats

In this sentence it is the direct object of hengjum, and the accusative plural form of this noun is also úlpurnar.

So:

• hengjum úlpurnar = hang the coats

Here are the basic forms you would usually look up:

• við = we
• setja = to put
• nýr = new
• motta = mat, rug
• í = in, into
• gangur = hallway, corridor
• og = and
• hengja = to hang
• úlpa = coat, parka
• á = on, onto
• snagi = peg, hook, coat rack/stand

This is useful because Icelandic words often appear in changed forms in real sentences.

Those endings include the suffixed definite article, which in English corresponds to the.

So:

• gangur = hallway
• ganginn = the hallway

and

• snagi = peg / hook / coat rack
• snagann = the peg / hook / coat rack

The exact form changes because of case and the noun’s declension pattern.

So in this sentence:

• í ganginn = into the hallway
• á snagann = onto the hook/rack

Very often, yes.

Instead of a separate word like English the, Icelandic usually attaches the article to the noun:

• motta = a mat

• mottan = the mat

• úlpur = coats

• úlpurnar = the coats

• gangur = hallway
• gangurinn = the hallway

But the exact form of the article changes with gender, number, and case, which is why you see forms like:

• ganginn
• snagann
• úlpurnar

This is a very important Icelandic pattern:

With motion toward a place/object:

use accusative

• í ganginn = into the hallway
• á snagann = onto the hook/rack

With location in/on a place/object:

use dative

• í ganginum = in the hallway
• á snaganum = on the hook/rack

So Icelandic often marks the difference between:

• movement to
• location at

with case.

Not necessarily. Snagi can mean:

• peg
• hook
• coat hook
• hanger point

Depending on context, á snagann could be translated naturally as:

• on the hook
• on the coat rack
• up on the peg

So even if the Icelandic word is singular, English may choose the most natural equivalent for the situation.

Yes, it is very normal.

The basic structure is:

• Við = subject
• setjum = verb
• nýja mottu = object
• í ganginn = prepositional phrase
• og hengjum = and hang
• úlpurnar = object
• á snagann = prepositional phrase

So the sentence flows in a straightforward way:

We put a new mat in the hallway and hang the coats on the hook/rack.

This is a very natural everyday Icelandic sentence structure.