Learn Hardcore Icelandic: Við hreinsum heimilið saman á laugardögum.

heimilið is the direct object, so it’s in the accusative singular.
The noun heimili is neuter; in neuter, nominative and accusative look the same.
The ending -ið is the suffixed definite article, so heimilið means the home.
Examples of other forms (just for orientation):
- nominative/accusative indefinite: heimili
- dative definite (e.g., in the home): á heimilinu

Why is it á laugardögum? What form is that?

For habitual actions on days of the week, Icelandic typically uses á
- dative plural: á laugardögum (on Saturdays, generally).
laugardagur (Saturday) declines to dative plural laugardögum. Rough pattern:
- singular: nominative laugardagur, accusative laugardag, dative laugardegi
- plural: nominative laugardagar, accusative laugardaga, dative laugardögum
For one specific Saturday, use the accusative singular:
- á laugardag or á laugardaginn (on Saturday, a one-off).

Could I say Þrífa instead of hreinsa here?

Yes. Þrífa is very common for cleaning a home and often sounds more natural in everyday speech.

Við þrífum heimilið saman á laugardögum.
Present tense of þrífa:
- ég þríf, þú þrífur, hann/hún/það þrífur, við þrífum, þið þrífið, þeir/þær/þau þrífa.
Nuance: hreinsa can mean clean/clear/purify (sometimes a bit “sanitize” or “clear out”), while þrífa is the default for physically cleaning the house.

Why is there no word for our in heimilið? Should it be heimilið okkar?

Icelandic often uses the definite article to imply possession when the possessor is obvious from context. Við hreinsum heimilið usually implies we clean our home.

To be explicit: Við hreinsum heimilið okkar (postposed possessive is the neutral, most common style).
You can also front it: Við hreinsum okkar heimili, which puts emphasis on our (not someone else’s).

Is heimilið the same as húsið or íbúðina?

Not exactly:

heimilið = the home (the household; can be a house, apartment, etc.)
húsið = the house (the building; neuter)
íbúðina = the apartment (accusative singular; feminine; nominative is íbúð) Choose based on what you mean.

Where can the time phrase go? Is the word order fixed?

Icelandic main clauses are verb-second (V2). Some natural options:

Neutral: Við hreinsum heimilið saman á laugardögum.
Fronted time for emphasis: Á laugardögum hreinsum við heimilið saman. If you front something (like the time phrase), the finite verb hreinsum must still be the second element.

Where should saman go? Can it move?

saman (together) is an adverb and is flexible:

Við hreinsum heimilið saman á laugardögum. (very natural)
Við hreinsum saman heimilið á laugardögum. (also fine)
Fronted for emphasis: Saman hreinsum við heimilið á laugardögum. (more rhetorical) Placing saman highlights togetherness; without it, the sentence doesn’t explicitly say you do it together (even though the subject is plural).

How do I say this if I mean we are cleaning right now?

Use the progressive construction vera + að + infinitive:

Við erum að hreinsa heimilið saman (núna). This describes an action in progress, unlike the simple present hreinsum, which often conveys a habit or general fact.

Is á always used for days? Are there alternatives?

For days of the week, á is standard: á laugardögum, á mánudögum, etc. Alternatives for expressing frequency:

á hverjum laugardegi (every Saturday; dative singular)
um helgar (at weekends)
adverbs like vikulega (weekly)

Why is there no article on laugardögum? Could I say á laugardögunum?

No article is used because it’s generic/habitual: on Saturdays in general. Á laugardögunum (with the article) would usually imply a specific known set of Saturdays and sounds odd in this habitual, general statement.

How do I negate the sentence?

Place ekki after the finite verb:

Við hreinsum ekki heimilið saman á laugardögum. With fronted time:
Á laugardögum hreinsum við ekki heimilið saman.

How do I turn it into a yes/no question? What about a wh-question?

Yes/no: move the verb first (V1): Hreinsum við heimilið saman á laugardögum?
Wh-question (e.g., when): Hvenær hreinsum við heimilið saman?

Any quick pronunciation tips for tricky parts?

Við: final ð is like the th in the English word this.
hreinsum: initial hr is a voiceless h+r cluster; ei is like the vowel in English say; stress on the first syllable.
heimilið: ei as above; final ð as in this.
laugardögum: au is like English oy (as in the word boy), ö is like French eu in deux; the g between vowels can soften; stress on the first syllable of each word: LAU-gar-dög-um.

Can I use a phrasal-looking form like hreinsa upp?

Yes, but it changes nuance:

hreinsa upp = clean up/clear out (often more thorough or about removing mess/clutter)
þrífa upp is also common for cleaning up. Your original sentence with that nuance: Við hreinsum upp/þrífum upp heimilið saman á laugardögum. (We clean up the home…)