Uppþvottavélin er hávær, svo ég loka hurðinni að eldhúsinu.

Yes. Icelandic usually expresses the by adding a definite ending to the noun.

• Base noun: uppþvottavél = dishwasher (feminine)
• Definite nominative singular: uppþvottavélin = the dishwasher
  (You’ll also see a separate article hinn/hin/hið, but the suffix is the normal everyday way.)

It’s a compound:

• upp- (up)
• þvottur (wash) → here in compound form þvott-
• vél (machine)
  So literally something like “washing-up machine” → “dishwasher”.

Adjectives agree with the noun in case, number, and gender, but some forms look identical across genders.

• hávær is nominative singular masculine and feminine
• The neuter would be hávært
  So Uppþvottavélin er hávær is fully agreeing (feminine nominative singular).

The comma separates two clauses, like English. svo here means so / therefore, introducing a result: “..., so I ...”.
In Icelandic, svo can also appear in meanings like then or in purpose constructions (depending on context), but in this sentence it’s the result linker.

Both can be grammatical, with slightly different feel.

• ..., svo ég loka hurðinni ... = “..., so I close the door ...” (subject-first; still normal)
• ..., svo loka ég hurðinni ... = “..., so I close the door ...” (more “then/so” fronted; common in narrative flow)
  In your sentence, ég loka is fine: it keeps a straightforward “I + verb” order.

Because loka (“to close”) typically takes a dative object in Icelandic.

• hurðin (nom) = the door
• hurðina (acc) = the door (accusative)
• hurðinni (dat) = the door (dative)
  So the verb’s government is: loka + dative → loka hurðinni.

hurðinni is:

• noun: hurð “door” (feminine)
• dative singular definite: -inni
  So it means to/for the door in form, even though the natural English translation is just “the door” after “close”.

In phrases like “the door to the kitchen,” Icelandic commonly uses að + dative to mean leading to / belonging to / connected with:

• hurðin að eldhúsinu = the door to the kitchen
  Here að governs the dative, so eldhúsið becomes eldhúsinu.

Yes, eldhúsinu is definite.

• Base: eldhús (neuter) = kitchen
• Definite nominative/accusative sg: eldhúsið = the kitchen
• Definite dative sg: eldhúsinu = (to/at) the kitchen
  So að eldhúsinu literally uses the dative “to/at the kitchen.”

Common ones here:

• þ in uppþvottavélin: like English th in thing
• ð in hurðinni: often like th in this, but can be very soft or disappear depending on position
• á in hávær: like “ow” in now (approx.)
• æ in hávær: like “eye” (approx.)
  And stress is usually on the first syllable: UPP-þvotta-vélin, ELD-húsinu.