Vettlingarnir liggja á stólnum við hurðina.

Because vettlingarnir is the definite form: the mittens / the gloves.

Breakdown:

• vettlingar = mittens/gloves
• -nir = the for a masculine nominative plural noun

So:

• vettlingar = mittens/gloves
• vettlingarnir = the mittens/gloves

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

Because the subject is plural.

• vettlingurinn liggur = the mitten/glove lies
• vettlingarnir liggja = the mittens/gloves lie

So the verb matches a plural subject here.

Icelandic very often prefers a position verb when describing where something is.

Here, liggja means something like:

• lie
• be lying
• be lying there

This is more natural than just using vera for many physical objects.

Common Icelandic location verbs include:

• liggja = lie, be lying
• standa = stand, be standing
• sitja = sit, be sitting

So for gloves resting on a chair, liggja is the natural choice.

Because á can take different cases depending on meaning.

Here it means on in the sense of location with no movement, so it takes the dative.

• stóll = chair
• stólnum = the chair in the dative singular

So:

• á stólnum = on the chair

That would usually imply movement onto the chair, not location on it.

A useful contrast:

• Vettlingarnir liggja á stólnum.
  The mittens/gloves are lying on the chair.
  → static location

• Ég set vettlingana á stólinn.
  I put the mittens/gloves onto the chair.
  → movement toward a destination

So with á:

• dative often = location
• accusative often = motion toward / onto

Because við in the sense of by / next to / near normally takes the accusative.

• hurð = door
• hurðina = the door in the accusative singular

So:

• við hurðina = by the door / next to the door

In the most natural reading, it describes the chair:

• á stólnum við hurðina
  = on the chair by the door

So the picture is:

• there is a chair
• the chair is by the door
• the mittens/gloves are on that chair

That is the default interpretation in this word order.

The dictionary forms are:

• vettlingur = mitten/glove
• liggja = to lie, be lying
• stóll = chair
• hurð = door

This is useful because Icelandic changes words a lot depending on case, number, and definiteness.

They show grammatical information such as definiteness, case, number, and gender.

In this sentence:

• vettlingar-nir

  • masculine
  • plural
  • nominative
  • definite
    → the mittens/gloves

• stól-num

  • masculine
  • singular
  • dative
  • definite
    → the chair

• hurð-ina

  • feminine
  • singular
  • accusative
  • definite
    → the door

So Icelandic often builds the directly onto the noun instead of using a separate word like English the.

Yes. Icelandic word order is fairly flexible, but the sentence as given is the most neutral and straightforward.

Normal order:

• Vettlingarnir liggja á stólnum við hurðina.

You could also front the location for emphasis:

• Á stólnum við hurðina liggja vettlingarnir.

That sounds more like:

• On the chair by the door lie the mittens/gloves

So the original sentence is the best basic pattern to learn first:

• subject + verb + place phrase

Yes—three things in particular:

1. Icelandic often uses position verbs

   • not just to be
   • here: liggja

2. Prepositions affect case

   • á stólnum = dative
   • við hurðina = accusative

3. The definite article is attached to the noun

   • not a separate word
   • vettlingarnir, stólnum, hurðina

So this one short sentence is a very good example of how Icelandic combines:

• noun endings
• case after prepositions
• and natural location verbs.