Sú sem vinnur á leikskólanum heilsar henni alltaf með brosi.

Sú means that one / the one in the feminine form. In this sentence, Sú sem... means the one who... referring to a female person.

So:

• Sú sem vinnur á leikskólanum = the woman/girl/person who works at the preschool

By contrast, hún usually just means she. If you used Hún sem..., that would not work here in standard Icelandic.

Sem is the relative particle who / that / which.

In this sentence, it connects Sú with the clause that describes her:

• Sú sem vinnur á leikskólanum
• literally: the one who works at the preschool

A useful pattern to remember is:

• sá sem = the one who (masculine)
• sú sem = the one who (feminine)
• það sem = what / that which (neuter, depending on context)

Unlike English relative pronouns, sem does not change form for case here.

Yes. The whole phrase acts like a noun phrase meaning the woman/person who works at the preschool.

Breakdown:

• Sú = the one (feminine)
• sem vinnur á leikskólanum = who works at the preschool

Together, they form the subject of the sentence.

So the structure is:

• [Subject] Sú sem vinnur á leikskólanum
• [Verb] heilsar
• [Object] henni
• [Adverb] alltaf
• [Prepositional phrase] með brosi

Vinna is the dictionary form, the infinitive, meaning to work.

Vinnur is the present tense, third person singular:

• ég vinn = I work
• þú vinnur = you work
• hann/hún/það vinnur = he/she/it works

Here the subject is Sú sem..., which is third person singular, so vinnur is the correct form.

Because á can take either the dative or the accusative, depending on meaning.

• á + dative = location, on/in/at
• á + accusative = motion toward, onto/to

Here the meaning is location: she works at the preschool, not movement toward it. So Icelandic uses the dative:

• á leikskólanum = at the preschool

Also note:

• leikskóli = preschool
• leikskólanum = the preschool in dative singular

The ending -num shows both definiteness and dative singular.

Because the verb heilsa takes the dative, not the accusative.

So:

• að heilsa einhverjum = to greet someone / say hello to someone

That means the pronoun must be dative:

• henni = to her / her (dative)

Compare:

• hana = accusative her
• henni = dative her

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

Yes, when it means to greet someone / say hello to someone, it normally governs the dative.

Examples:

• Ég heilsa honum. = I greet him.
• Hún heilsaði mér. = She greeted me.
• Við heilsum þeim. = We greet them.

So in your sentence:

• heilsar henni = greets her

A good habit in Icelandic is to learn verbs together with the case they require:

• heilsa + dative

Alltaf means always.

In this sentence it modifies the verb heilsar, so:

• heilsar henni alltaf = always greets her

Its placement is very natural in Icelandic. Adverbs like alltaf often appear after the object or around the verb phrase, depending on structure and emphasis.

This sentence sounds normal as given:

• Sú sem vinnur á leikskólanum heilsar henni alltaf með brosi.

If you move alltaf, the sentence may still be grammatical, but the emphasis can change.

Because the preposition með takes the dative in this meaning.

So:

• bros = a smile (nominative/accusative form)
• brosi = smile (dative singular)

Therefore:

• með brosi = with a smile

This is another common Icelandic pattern:

• með + dative

Examples:

• með vini = with a friend
• með barninu = with the child
• með brosi = with a smile

It is a prepositional phrase describing how she greets her.

So:

• heilsar henni = greets her
• alltaf = always
• með brosi = with a smile

In other words, it tells you the manner of the greeting.

It can definitely be broken down:

• leikskóli = preschool
• leikskól- = stem
• -anum / -num = definite article + case ending, depending on the form

In this sentence, leikskólanum is:

• singular
• definite = the preschool
• dative = because of á with location

So á leikskólanum literally means at the preschool.

Not very naturally. A rough word-for-word version would be:

• Sú = that one / the one
• sem = who
• vinnur = works
• á leikskólanum = at the preschool
• heilsar = greets
• henni = her (dative)
• alltaf = always
• með brosi = with a smile

But natural English usually needs some adjustment, because Icelandic uses cases and structures that English does not.

So while a literal breakdown helps you understand the grammar, it is better not to expect a perfect word-for-word match.

Yes. Instead of Sú sem..., Icelandic could use a specific noun phrase, for example a word meaning the woman or the teacher, depending on context.

But Sú sem... is useful when you want to say the one who... without naming the person directly.

It is a very common Icelandic pattern for referring to someone by description rather than by name or noun.