Þau fluttu inn eftir því hvort heimilið var tilbúið.

Yes. Þau is the nominative plural neuter form of they.

In Icelandic, þau is used:

• for a mixed-gender group
• for children
• for a group whose gender is unspecified or not important

So it does not necessarily mean that the people are literally neuter. It is often just the normal pronoun for a mixed group.

Compare:

• þeir = they (all male, or masculine default in some contexts)
• þær = they (all female)
• þau = they (mixed group / children / unspecified)

Fluttu is the 3rd person plural past tense of flytja.

So:

• flytja = to move, transport
• flutti = moved (singular)
• fluttu = moved (plural)

Because the subject is þau = they, the verb has to be plural:

• Þau fluttu = They moved

Yes. Here flytja inn works very much like English move in.

• flytja = move / transport
• inn = in, inward

Together, in this context:

• flytja inn = to move in

So Þau fluttu inn means They moved in.

A useful note: Icelandic often uses a verb + directional adverb combination like this:

• fara inn = go in
• koma út = come out
• flytja inn = move in

Also, flytja inn can mean import in other contexts, so context matters.

Here eftir því means something like depending on that or according to that.

In this sentence, it leads into a clause with hvort, so the whole structure means:

• eftir því hvort ... = depending on whether ...

So the phrase is not meant to be translated word-for-word too mechanically. It is better understood as a set expression.

Because eftir takes the dative in this construction, and the dative form of það is því.

So:

• nominative: það
• dative: því

That is why Icelandic says:

• eftir því
  not
• eftir það

This is a very common thing in Icelandic: prepositions often control a particular case.

Because hvort means whether, while ef more often means if in a conditional sense.

Here the idea is not really a straightforward condition like:

• If the home was ready, they moved in

Instead, the sentence presents a whether-or-not idea:

• depending on whether the home was ready

That is why hvort is the natural choice.

A simple contrast:

• ef = if
• hvort = whether

Yes. Heimilið comes from heimili, which usually means:

• home
• household
• place of residence

It is a little broader and warmer than just the physical building.

Compare:

• heimili = home / household
• hús = house / building

So heimilið var tilbúið suggests the home was ready, not just the building existed.

Because tilbúið has to agree with heimilið, which is neuter singular.

The adjective tilbúinn changes form depending on gender and number:

• masculine singular: tilbúinn
• feminine singular: tilbúin
• neuter singular: tilbúið

Since:

• heimilið is neuter singular

you get:

• heimilið var tilbúið = the home was ready

It is in the nominative singular.

That is because heimilið is the subject of the subordinate clause:

• hvort heimilið var tilbúið

Inside that clause:

• heimilið = subject
• var = was
• tilbúið = ready

Also, the ending -ið is the definite article attached to the noun, so heimilið means the home.

Because this is a subordinate clause, and Icelandic usually does not use main-clause inversion there.

In a main clause, Icelandic often follows verb-second word order. But after a subordinating word like hvort, the clause usually has more regular subject-verb order:

• hvort heimilið var tilbúið

That is the normal pattern:

• hvort
  • subject
    • verb
      • complement

So the structure is completely natural.