Sameiginlega þvottahúsið sparar okkur tíma, en sameiginlegi þurrkarinn í kjallaranum er sjaldan notaður.

Because the adjective has to agree with the noun in gender, number, and case.

• þvottahúsið is neuter singular nominative
• þurrkarinn is masculine singular nominative

Also, both nouns are definite (the laundry room, the dryer), so the adjective takes the weak declension:

• sameiginlega þvottahúsið = weak, neuter, singular, nominative
• sameiginlegi þurrkarinn = weak, masculine, singular, nominative

So the different adjective endings are normal agreement endings.

Because Icelandic usually uses the weak adjective when the noun is definite.

Compare:

• sameiginlegt þvottahús = a shared laundry room
• sameiginlega þvottahúsið = the shared laundry room

So:

• sameiginlegt = strong adjective form, often with indefinite nouns
• sameiginlega = weak adjective form, used here because þvottahúsið is definite

This is one of the most important adjective patterns in Icelandic.

No. Here it is an adjective, because it describes the noun þvottahúsið.

That can be confusing because sameiginlega can also look like an adverb form. But in this sentence, it is clearly modifying a noun, so it is adjectival:

• sameiginlega þvottahúsið = the shared laundry room

If it were an adverb, it would modify a verb, adjective, or whole clause instead.

Those endings are the suffixed definite article in Icelandic. Instead of a separate word like English the, Icelandic usually attaches it to the noun.

In this sentence:

• þvottahúsið = þvottahús + ið = the laundry room
• þurrkarinn = þurrkari + inn = the dryer
• kjallaranum = kjallari + anum/num-type dative definite ending = the basement in the dative

So Icelandic often expresses the as part of the noun itself.

Okkur is dative plural of við (we/us).

It appears because the verb spara often works like this:

• spara einhverjum eitthvað = save someone something

So in:

• sparar okkur tíma

the roles are:

• okkur = to us / for us → dative
• tíma = time → accusative direct object

A natural way to think of it is: the shared laundry room saves us time.

Because tíma is the accusative singular form of tími.

The noun tími is the direct object of sparar, so it appears in the accusative:

• nominative: tími
• accusative: tíma

So:

• þvottahúsið sparar okkur tíma

uses tíma because it is what is being saved.

Sparar is the 3rd person singular present tense of spara.

It is singular because the subject is singular:

• Sameiginlega þvottahúsið = one laundry room

So Icelandic uses:

• þvottahúsið sparar = the laundry room saves

If the subject were plural, the verb form would change accordingly.

Notaður is the past participle of nota (to use), and here it is used with er to make a passive-style meaning:

• er notaður = is used

The ending -ur is there because it agrees with þurrkarinn, which is:

• masculine
• singular
• nominative

So the participle matches the noun:

• þurrkarinn er notaður = the dryer is used

If the noun were feminine or neuter, the participle form would be different.

Because that is a very normal Icelandic word order for adverbs.

In this clause:

• er sjaldan notaður

the adverb sjaldan (rarely) comes after the finite verb er and before the participle notaður.

This is similar to English:

• is rarely used

So the placement is not unusual; it is standard Icelandic syntax.

Because í takes different cases depending on meaning:

• location = usually dative
• movement into = usually accusative

Here the meaning is location:

• í kjallaranum = in the basement

So kjallaranum is in the dative singular definite form.

Compare the general idea:

• í kjallaranum = in the basement (location)
• fara í kjallarann = go into the basement (movement)

Yes. Icelandic uses compound nouns very often.

• þvottahús = þvotta + hús
  Literally something like washing-house, meaning laundry room
• þurrkari comes from the idea of drying/dryer

A useful rule is that in Icelandic compounds, the last part usually determines the gender:

• hús is neuter, so þvottahús is neuter
• nouns ending in -ari are very often masculine, so þurrkari is masculine

That helps explain why the sentence has:

• þvottahúsið as neuter
• þurrkarinn as masculine

En means but and connects the two clauses:

• Sameiginlega þvottahúsið sparar okkur tíma
• en sameiginlegi þurrkarinn í kjallaranum er sjaldan notaður

So it introduces a contrast:

• the shared laundry room is useful in one way,
• but the shared dryer is rarely used.

It is a very common coordinating conjunction in Icelandic.