Í desember erum við mest heima.

Because Icelandic normally follows a verb-second pattern in main clauses.

That means:

• the first slot can be a time phrase, place phrase, object, etc.
• but the finite verb still comes in the second slot

So when Í desember is placed first, the verb erum must come next:

• Í desember erum við mest heima.

If you start with the subject instead, you get:

• Við erum mest heima í desember.

Both are grammatical, but the first one puts extra focus on December.

Erum is the 1st person plural present tense form of vera, which means to be.

So:

• við erum = we are

Useful present-tense forms of vera are:

• ég er = I am
• þú ert = you are
• hann / hún / það er = he / she / it is
• við erum = we are
• þið eruð = you are (plural)
• þeir / þær / þau eru = they are

Here mest means most or the most.

It is the superlative form used adverbially, so it tells you that this is the time when something happens more than at other times.

So við erum mest heima means:

• we are at home the most
• or more naturally in English, we are home the most
• sometimes even we are mostly at home, depending on context

It implies a comparison with other times of year.

Because heima means at home (location), while heim means home / homewards (direction).

Compare:

• Ég er heima. = I am at home.
• Ég fer heim. = I go home.

In your sentence, the verb is vera (to be), which describes location, not movement, so heima is the correct choice.

No. In this sentence, heima is an adverb, not a noun.

It describes where someone is:

• heima = at home

That is why it does not change for gender, number, or case here.

This is different from English, where home can look like a noun but also behaves adverbially in phrases like stay home or go home.

Yes, absolutely.

Both are natural:

• Í desember erum við mest heima.
• Við erum mest heima í desember.

The difference is mainly about focus:

• Í desember... puts the time frame first and highlights December
• Við erum... starts more neutrally with the subject we

This kind of fronting is very common in Icelandic.

Not here. In this sentence, mest is being used as an adverb, so it does not agree with við or with heima.

So it just stays:

• mest

You do not change it for gender, number, or case in this use.

Í desember means in December.

A useful thing to remember is that month names in Icelandic are normally written with lower-case letters, not capitals, unless they begin the sentence.

So:

• í desember = in December
• í janúar = in January
• í ágúst = in August

That is why desember is not capitalized here.

No, not normally.

In Icelandic, vera heima usually means to be at home physically. So erum við mest heima is understood as:

• we are at home the most
• we spend the most time at home

It is about where you are, not about feeling emotionally comfortable or relaxed.

Yes. Very common.

Icelandic often uses vera heima literally be at home where English might say either:

• be at home
• stay home
• stay at home

So Í desember erum við mest heima can be understood naturally as:

• In December we are at home the most
• In December we stay home the most

Both capture the idea well.