Á brottfarardaginn athugum við rotþróna eftir vatnsleysið og lokum sumarbústaðnum.

Because Icelandic main clauses normally follow the verb-second (V2) rule.

Here, the sentence starts with the time phrase Á brottfarardaginn. Once that element is placed first, the finite verb comes next:

• Á brottfarardaginn athugum við ...

If there were no fronted time phrase, you would normally say:

• Við athugum ...

So this word order is not unusual at all; it is standard Icelandic syntax.

Á brottfarardaginn is a time expression meaning something like on the departure day.

The preposition á is often used with the accusative in fixed expressions for specific days or points in time, for example:

• á mánudaginn
• á jóladag
• á brottfarardaginn

So brottfarardagur becomes brottfarardaginn, which is the definite accusative singular form.

Because Icelandic forms compounds very freely, and they are normally written as one word.

brottfarardaginn can be broken down like this:

• brottför = departure
• brottfarar- = a compound form based on the genitive
• dagur = day
• -inn = the definite article the

So the whole thing is literally a compound meaning the departure-day.

This is very typical Icelandic style: ideas that English often writes as several words are often packed into a single compound noun in Icelandic.

Both are 1st person plural, present indicative forms:

• athugum = we check / we inspect
• lokum = we close / we shut

Their dictionary forms are:

• athuga
• loka

The ending -um is a common present-tense ending for we.

Because rotþróna is the definite singular accusative form of rotþró.

In this sentence, it is the object of athuga, and athuga normally takes the accusative:

• athuga rotþró = inspect a septic tank
• athuga rotþróna = inspect the septic tank

So the ending shows that the noun is:

• definite: the
• singular
• accusative

Here eftir means after in a time sense, and in that meaning it takes the accusative.

So:

• vatnsleysi = water shortage / lack of water
• vatnsleysið = the water shortage / the lack of water

Because vatnsleysi is a neuter noun, its nominative and accusative singular definite forms are the same, so vatnsleysið looks the same in both cases.

A useful thing to remember is that eftir can take different cases depending on meaning. In this sentence, because it means after, it takes the accusative.

Because Icelandic compounds often use the genitive form of the first element.

Here:

• vatn = water
• vatns = of water
• leysi = lack / absence

So vatnsleysi literally means something like lack of water.

This same pattern appears in many Icelandic compounds. It is also similar to what happens in brottfarardagur, where the first part is also in a compound form based on the genitive.

Because the verb loka takes a dative object in Icelandic.

So even though English says close the summer house, Icelandic uses the dative:

• loka sumarbústaðnum

This is just something you have to learn with the verb. Many Icelandic verbs govern a particular case that does not match English.

For example:

• loka hurðinni = close the door
• loka bílnum = lock/close the car

So in your sentence, sumarbústaðnum is correct because loka requires the dative.

Because both verbs share the same subject.

The structure is:

• athugum við ... og lokum ...

This means:

• we inspect ... and close ...

Once við has already been stated, Icelandic does not need to repeat it before the second verb unless there is some special emphasis or contrast.

So this is the normal, natural way to say it.

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

In this sentence you can see several examples:

• brottfarardaginn = brottfarardagur
  • definite ending, accusative singular masculine
• rotþróna = rotþró
  • definite ending, accusative singular feminine
• vatnsleysið = vatnsleysi
  • definite ending, accusative singular neuter
• sumarbústaðnum = sumarbústaður
  • definite ending, dative singular masculine

So the ending does two jobs at once:

• it shows definiteness (the)
• it also reflects the noun’s case and number

That is why the endings are different from noun to noun.

Yes. That would also be grammatical.

Compare:

• Við athugum rotþróna eftir vatnsleysið og lokum sumarbústaðnum.
• Á brottfarardaginn athugum við rotþróna eftir vatnsleysið og lokum sumarbústaðnum.

The version with Á brottfarardaginn first puts extra focus on when this happens. Once that time phrase is moved to the front, the verb shifts into second position, giving athugum við.

So the difference is mainly about word order and emphasis, not basic meaning.