Heimasjúkraþjálfunin byrjar í næstu viku.

Because Icelandic very often forms compound nouns as a single word.

Here, heimasjúkraþjálfunin is built from:

• heima- = home-, at-home
• sjúkraþjálfun = physiotherapy
• -in = the definite article

So Icelandic packs the whole idea into one word where English would usually use several words.

It is the suffixed definite article, basically the.

So:

• heimasjúkraþjálfun = home physiotherapy
• heimasjúkraþjálfunin = the home physiotherapy

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

The dictionary form is heimasjúkraþjálfun.

The -in is just the definite ending, so if you look the word up, you would normally search for heimasjúkraþjálfun.

Its grammatical gender is feminine, because the final part sjúkraþjálfun is feminine, and compound nouns normally take the gender of the last element.

Because byrjar is the present tense form used with a third-person singular subject.

The verb is byrja = to begin, to start.

In the present tense:

• ég byrja = I begin
• þú byrjar = you begin
• hún/hann/það byrjar = she/he/it begins

Here the subject is heimasjúkraþjálfunin, which is singular, so byrjar is the correct form.

This is the normal way to say next week in this kind of sentence.

• í = in
• vika = week
• í næstu viku = in the next week / next week

Icelandic often uses í + dative for time expressions like this.

Because the phrase is in the dative case after í.

The base forms are:

• næstur = next
• vika = week

But in í næstu viku, both words change form:

• næstu = dative singular feminine
• viku = dative singular of vika

So the grammar of the whole phrase determines the form, not just the dictionary form.

Words like næsti (next), síðasti (last), and þessi (this) often behave in a definite way in Icelandic, so they commonly take weak adjective endings.

That is why you get:

• næsta vika = next week
• í næstu viku = next week / in the next week

So this is not an unusual special case; it is the normal pattern.

Yes.

You can say:

• Í næstu viku byrjar heimasjúkraþjálfunin.

That is very natural Icelandic.

But remember the Icelandic verb-second rule: when you move something to the front, the finite verb usually stays in second position. So:

• correct: Í næstu viku byrjar heimasjúkraþjálfunin
• not normal: Í næstu viku heimasjúkraþjálfunin byrjar

Yes, but the meaning changes slightly.

• Heimasjúkraþjálfunin byrjar í næstu viku. = The home physiotherapy starts next week.
• Heimasjúkraþjálfun byrjar í næstu viku. = Home physiotherapy starts next week.

The version with -in sounds more specific, as if the listener already knows which treatment or service is being talked about.

A rough learner-friendly guide is:

• Heimasjúkraþjálfunin ≈ HAY-ma-syoo-kra-thyaul-fu-nin
• byrjar ≈ BIR-yar
• í ≈ ee
• næstu ≈ NIGH-stu
• viku ≈ VEE-ku

A few useful pronunciation points:

• Stress usually falls on the first syllable of each word.
• þ sounds like th in thing.
• j sounds like y in yes.
• æ is roughly like i in fine.
• í is like ee in see.

The exact sounds are a bit different from English, but this will get you close.

Usually it refers to the treatment arrangement, service, or course of therapy rather than one single appointment.

So the sentence most naturally suggests something like:

• the home physiotherapy program starts next week
• the home physiotherapy service begins next week

If you wanted to emphasize a single session, the wording might be different, depending on context.

It is:

subject + verb + time phrase

• Heimasjúkraþjálfunin = subject
• byrjar = verb
• í næstu viku = time expression

So the sentence structure is actually quite close to English in this example, even though the grammar inside the words is more inflected.