Eftir nokkrar vikur venst hún betur lífinu í nýju hverfi.

Because Icelandic is a verb-second language. When the sentence starts with something other than the subject, here Eftir nokkrar vikur, the finite verb usually comes next.

So the structure is:

• Eftir nokkrar vikur = first element
• venst = finite verb in second position
• hún = subject

If you started with the subject instead, you could also say:

• Hún venst betur lífinu í nýju hverfi eftir nokkrar vikur.

That version is also grammatical, but it puts less emphasis on the time phrase.

Venst comes from the verb venjast, which means to get used to or to become accustomed to.

So in this sentence:

• venst = gets used to / becomes accustomed to

A useful comparison is:

• venja = to accustom someone to something
• venjast = to get used to something oneself

So venjast is the form you use when the subject is the one adapting.

In English, we say get used to something, with the preposition to. Icelandic often expresses that same relationship through case instead of a separate preposition.

The verb venjast takes a dative object, so the idea of English to is built into the grammar of the noun phrase itself.

That is why you get:

• lífinu = to the life / life in the sense required by venjast

So you do not add another word corresponding to English to.

Because venjast takes the dative case, and lífinu is the dative singular definite form of líf.

Breakdown:

• líf = life
• lífi = dative singular
• lífinu = dative singular with the definite article, the life

So lífinu is there because of two things at once:

1. the verb requires dative
2. the noun is definite

Because betur is an adverb, and it modifies the verb venst.

• vel = well
• betur = better

In this sentence, it describes how she gets used to life, not what kind of life it is. So Icelandic uses the adverb betur, not an adjective.

A natural sense here is:

• she adjusts better
• she gets used to it more easily
• she settles in better

Here eftir means after, and with this time meaning it takes the accusative case.

So:

• vika = week
• vikur = accusative plural
• nokkrar = some / a few, agreeing with vikur

Because vika is a feminine noun, nokkrar also has to be feminine plural accusative.

So eftir nokkrar vikur means after a few weeks.

Because the preposition í can take either:

• accusative for motion into something
• dative for location or being in something

Here there is no movement into the neighborhood. The phrase describes life in a new neighborhood, so dative is used.

Breakdown:

• hverfi = neighborhood, district
• nýju = the adjective new in dative singular neuter
• hverfi = dative singular neuter form here

So í nýju hverfi means in a new neighborhood.

Icelandic uses a suffixed definite article, attached to the noun itself.

So:

• lífinu = the life in dative singular
• hverfi = a neighborhood / neighborhood without the definite article

This sentence treats life as a specific thing she is getting used to, but new neighborhood as an indefinite setting.

That combination is completely normal in Icelandic:

• definite noun in one phrase
• indefinite noun in another

Yes. A version with the subject first is also possible:

• Hún venst betur lífinu í nýju hverfi eftir nokkrar vikur.

That is grammatically fine. The difference is mainly information structure:

• Eftir nokkrar vikur venst hún... highlights the time frame first
• Hún venst... eftir nokkrar vikur starts more neutrally with the subject

So the original sentence is a very typical Icelandic way to emphasize when this happens.