Ég vil lifa rólegu lífi í sveitinni.

Vil is the 1st person singular present form of vilja (to want).

So:

• ég vil = I want
• þú vilt = you want
• hann/hún/það vill = he/she/it wants

In this sentence, Ég vil... is the normal way to say I want....

Because it comes after vil.

Just like English says I want to live, Icelandic uses vil + infinitive:

• Ég vil lifa = I want to live

Only vil is conjugated. Lifa stays in the basic dictionary form.

Because lifa and búa are not quite the same.

• lifa = to live, in the broader sense of living life
• búa = to live/reside, meaning where you live

So:

• Ég vil lifa rólegu lífi = I want to live a quiet life
• Ég vil búa í sveitinni = I want to live in the countryside

The sentence you were given focuses on the kind of life the speaker wants, not just their address.

Because in the common Icelandic expression lifa ... lífi (live a ... life), the noun líf appears in the dative:

• lifa góðu lífi = live a good life
• lifa rólegu lífi = live a quiet life

So this is not just the plain nominative rólegt líf. The phrase requires the dative form:

• líf → lífi
• rólegt → rólegu

This is one of those patterns that learners usually memorize as a set expression.

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

Here lífi is:

• neuter
• singular
• dative

So rólegur changes to the matching form rólegu.

A useful way to think of it is:

• the noun changes form
• the adjective changes with it

Sveitinni is dative singular definite.

That happens because the preposition í can take two cases:

• accusative for motion into
• dative for location in

Here the meaning is location, not movement:

• í sveitinni = in the countryside

Compare:

• Ég er í sveitinni = I am in the countryside → location → dative
• Ég fer í sveitina = I am going into/to the countryside → motion → accusative

It includes the definite article.

In Icelandic, the is usually attached to the end of the noun instead of being a separate word.

So:

• sveit = countryside / rural district
• sveitinni = the countryside in the dative singular

The ending looks longer because both the case ending and the definite article are built into the word.

Yes, sveit often means the countryside or a rural area.

It can also have more specific meanings depending on context, such as:

• a rural district
• a parish-like local area
• the country, as opposed to the city

In this sentence, í sveitinni is very naturally understood as in the countryside.

Usually no. Icelandic normally does include the subject pronoun.

So Ég vil lifa... is the normal form, not just Vil lifa...

Unlike strongly pro-drop languages such as Spanish or Italian, Icelandic usually keeps pronouns like:

• ég = I
• þú = you
• hann/hún = he/she

Leaving them out would usually sound incomplete except in very special contexts.

Yes: it follows the normal Icelandic main-clause pattern, where the finite verb is in second position.

Here that looks like:

• Ég = first element
• vil = finite verb, second position
• lifa rólegu lífi í sveitinni = the rest

This is the usual pattern in straightforward statements.

You can also see the verb-second rule if something else comes first:

• Í sveitinni vil ég lifa rólegu lífi.

Even then, vil still stays in second position.

Because Icelandic often expresses indefiniteness without a separate word like English a/an.

So:

• rólegu lífi can mean a quiet life

Icelandic has a definite article built onto nouns, but it usually does not have an indefinite article corresponding to English a/an. The noun phrase is simply left indefinite.