Heimasjúkraþjálfunin, dagdvölin og stuðningsþjónustan hjálpa skjólstæðingnum.

Because the subject is plural.

The sentence has three things as the subject:

• Heimasjúkraþjálfunin
• dagdvölin
• og stuðningsþjónustan

Even though each noun is singular by itself, together they form a plural subject: X, Y and Z help...

So the verb hjálpa is the present plural form of að hjálpa.

A useful contrast:

• Heimasjúkraþjálfunin hjálpar ... = The home physiotherapy helps ...
• Heimasjúkraþjálfunin, dagdvölin og stuðningsþjónustan hjálpa ... = The home physiotherapy, day care, and support service help ...

Because að hjálpa takes the dative case in Icelandic.

So the person receiving the help is not in the accusative, as an English speaker might expect, but in the dative.

• skjólstæðingur = a client / recipient of services
• skjólstæðingnum = the client in the dative singular definite

You can think of it like this:

• að hjálpa einhverjum = to help someone

So:

• hjálpa skjólstæðingi = help a client
• hjálpa skjólstæðingnum = help the client

This is one of the most important things to memorize with Icelandic verbs: many verbs require a specific case.

Yes. They are the subject of the sentence, so they are in the nominative case.

They are the things doing the helping:

• Heimasjúkraþjálfunin
• dagdvölin
• stuðningsþjónustan

So the basic structure is:

• [nominative subject] + [verb] + [dative object/complement]

In this sentence:

• Heimasjúkraþjálfunin, dagdvölin og stuðningsþjónustan = nominative subject
• hjálpa = verb
• skjólstæðingnum = dative

Those endings show the definite article, which in Icelandic is usually attached to the end of the noun instead of standing as a separate word like English the.

So:

• heimasjúkraþjálfun = home physiotherapy

• heimasjúkraþjálfunin = the home physiotherapy

• dagdvöl = day care / day stay

• dagdvölin = the day care / the day program

• stuðningsþjónusta = support service
• stuðningsþjónustan = the support service

The exact ending changes depending on the noun’s gender and form, so you do not always get the same definite ending.

In Icelandic, each noun normally carries its own definiteness if each one is meant as a specific thing.

So:

• Heimasjúkraþjálfunin, dagdvölin og stuðningsþjónustan

means that all three are definite, specific services.

This is very natural in Icelandic. You would not normally leave the article off the second and third nouns if all three are definite.

It works much like English repeating the:

• the home physiotherapy, the day care, and the support service

Even if English often sounds smoother without repeating the, Icelandic usually marks each definite noun directly.

Icelandic uses compounds very freely, and this sentence has several good examples.

1. Heimasjúkraþjálfunin

This is built from:

• heima = at home
• sjúkra = related to the sick / patients
• þjálfun = training, therapy, rehabilitation

Together, heimasjúkraþjálfun means something like home physiotherapy.

2. dagdvölin

This comes from:

• dagur / dag- = day
• dvöl = stay, sojourn

So dagdvöl is a day stay, often used for a day-care or day-program service.

3. stuðningsþjónustan

This is:

• stuðningur / stuðnings- = support
• þjónusta = service

So stuðningsþjónusta = support service.

Compounds are very common in Icelandic, so learning to spot the smaller parts inside a long word is extremely helpful.

Yes, gender matters because it affects noun forms, especially the definite article and other inflections.

In this sentence:

• heimasjúkraþjálfun is feminine
• dagdvöl is feminine
• stuðningsþjónusta is feminine
• skjólstæðingur is masculine

That helps explain forms like:

• heimasjúkraþjálfunin
• dagdvölin
• stuðningsþjónustan

These are feminine singular nominative definite forms.

And:

• skjólstæðingnum

is masculine singular dative definite.

So yes, gender is definitely part of why the words look the way they do.

The dictionary form is skjólstæðingur.

That is the nominative singular indefinite form, which is the form dictionaries usually list.

Here is the rough pattern:

• skjólstæðingur = a client
• skjólstæðing = a client (accusative)
• skjólstæðingi = a client (dative)
• skjólstæðings = of a client (genitive)

With the definite article:

• skjólstæðingnum = the client (dative)

Since hjálpa takes dative, the sentence uses skjólstæðingnum.

This is basically normal Icelandic word order.

The sentence follows a straightforward pattern:

• subject
  • verb
    • object/complement

So:

• Heimasjúkraþjálfunin, dagdvölin og stuðningsþjónustan
• hjálpa
• skjólstæðingnum

That is a very neutral way to say it.

Icelandic does have verb-second behavior, so in other sentence types you may see the verb move earlier if something else comes first. But in this sentence, nothing unusual is happening.

Yes, but it would mean something slightly different.

The indefinite version would be:

• Heimasjúkraþjálfun, dagdvöl og stuðningsþjónusta hjálpa skjólstæðingi.

That would mean something more like:

• Home physiotherapy, day care, and support services help a client / clients

It sounds more general, less tied to specific known services or a specific known client.

The original sentence with definite forms:

• Heimasjúkraþjálfunin, dagdvölin og stuðningsþjónustan hjálpa skjólstæðingnum.

refers to specific services helping the specific client.

This is a punctuation choice used when listing several coordinated items, especially in more formal or administrative writing.

The sentence lists three things:

• Heimasjúkraþjálfunin
• dagdvölin
• og stuðningsþjónustan

The comma helps separate the items clearly.

You may notice some variation in punctuation style depending on the text type, but in formal Icelandic prose, this kind of comma in a list is completely normal.

The important grammar point is that all three nouns belong together as the subject, and og links the final item in the list.