Learn Hardcore Icelandic: Hún hefur mikla þolinmæði þegar hún útskýrir erfiða hluti fyrir börnunum. - She has a lot of patience when she explains difficult things to the children.

þolinmæði is a feminine singular noun.
In this sentence, þolinmæði is the direct object of hefur, so it is in the accusative case.
The adjective mikill (much / a lot) in the feminine accusative singular form is mikla.

Very simplified:

Nominative feminine: mikil þolinmæði (e.g. as subject)
Accusative feminine: mikla þolinmæði (as direct object, like here)

So mikla þolinmæði is required by the grammar (feminine + singular + accusative).

Is hefur here an auxiliary (like an English perfect) or just the verb to have?

Here hefur is a main verb meaning has / possesses.

As an auxiliary in the present perfect, hefur would be followed by a past participle, for example:
- Hún hefur haft mikla þolinmæði – She has had a lot of patience.

In your sentence there is no past participle, only the noun þolinmæði, so we are not in a perfect tense; it is simply:

Hún hefur mikla þolinmæði – She has a lot of patience.

Could I also say Hún er mjög þolinmóð þegar hún útskýrir erfiða hluti fyrir börnunum? What is the difference?

Yes, that is correct Icelandic, but there is a nuance difference:

Hún hefur mikla þolinmæði – literally She has a lot of patience.
- Uses the noun þolinmæði.
- Focuses slightly more on the amount of patience.
Hún er mjög þolinmóð – She is very patient.
- Uses the adjective þolinmóður / þolinmóð / þolinmótt (here feminine þolinmóð).
- Describes her character/quality.

In context, both sentences are natural and often close in meaning; hafa mikla þolinmæði sounds a bit more like talking about her patience in that particular situation, while vera mjög þolinmóð can sound more like a general trait, though context can blur this difference.

Why is it erfiða hluti and not something like erfiðir hlutir?

Because erfiða hluti is in the accusative plural, not the nominative.

hlutur = thing (masculine)
- Nominative plural: hlutir
- Accusative plural: hluti
erfiður = difficult
- Nominative plural masculine: erfiðir
- Accusative plural masculine: erfiða

The verb útskýra (to explain) takes a direct object in the accusative case, so in the phrase að útskýra erfiða hluti:

Noun: hluti – accusative plural
Adjective: erfiða – accusative plural masculine to match hluti

If erfiða hluti were the subject, you’d see nominative: erfiðir hlutir.

Why is börnunum in the dative case after fyrir?

The preposition fyrir can take either accusative or dative, depending on meaning and nuance. Here it takes the dative.

With benefit / on behalf of / for someone’s sake, fyrir usually takes dative.
- fyrir börnunum – for the children (for their benefit).
With some other meanings (like movement in front of something, “before”, “ago”), it often takes accusative.

In this sentence, the meaning is clearly for the children, so börnunum is dative plural of börn:

Indefinite dative plural: börnum
Definite dative plural: börnunum

How do we get the form börnunum from the word barn?

Barn (child) is neuter and has an irregular plural börn. The plural declension is:

Indefinite plural:
- Nominative: börn
- Accusative: börn
- Dative: börnum
- Genitive: barna

To make the forms definite, Icelandic adds a suffix to the noun:

Nominative definite: börnin – the children
Accusative definite: börnin
Dative definite: börnunum
Genitive definite: barnanna

So börnunum = börnum (dative plural) + definite ending. It means to/for the children.

Is the word order hún útskýrir erfiða hluti fyrir börnunum fixed, or can fyrir börnunum move?

Icelandic word order is fairly flexible, but there are preferences:

All of these are grammatical:

Hún útskýrir erfiða hluti fyrir börnunum.
Hún útskýrir fyrir börnunum erfiða hluti.

The usual neutral tendency is:

Subject – Verb – (Indirect object) – (Direct object) – Other phrases

So with fyrir börnunum (a kind of indirect object), many speakers find:

Hún útskýrir fyrir börnunum erfiða hluti

slightly more neutral, while

Hún útskýrir erfiða hluti fyrir börnunum

can sound like it emphasizes erfiða hluti a bit more (what she explains).

In your sentence introduced by þegar, the same applies inside the clause:

þegar hún útskýrir erfiða hluti fyrir börnunum
þegar hún útskýrir fyrir börnunum erfiða hluti

Both are fine.

Why is it þegar hún útskýrir and not þegar hún er að útskýra?

Both are possible; they differ in aspect and nuance:

þegar hún útskýrir – literally when she explains.
- Simple present, which in Icelandic can cover both English when she explains and when she is explaining depending on context.
þegar hún er að útskýra – literally when she is (in the process of) explaining.
- vera + að + infinitive is a common way in Icelandic to mark a more explicit ongoing action, similar to the English be doing form.

In your sentence:

Hún hefur mikla þolinmæði þegar hún útskýrir erfiða hluti fyrir börnunum.
sounds perfectly natural as a general statement.

Using er að útskýra adds a bit more focus on the ongoing process, but is not necessary.

Could you drop the second hún and say Hún hefur mikla þolinmæði þegar útskýrir erfiða hluti fyrir börnunum?

No, that would be ungrammatical in standard Icelandic.

Unlike some languages, Icelandic does not normally allow you to drop the subject pronoun in a clause like this. Each finite clause needs its own subject:

Hún hefur mikla þolinmæði þegar hún útskýrir erfiða hluti fyrir börnunum.

You need to repeat hún in the þegar-clause, even though the subject is the same as in the main clause.

Is þolinmæði treated as a countable or uncountable noun in Icelandic?

þolinmæði is generally treated as an uncountable (mass) noun, similar to English patience:

mikil / mikla þolinmæði – a lot of patience
You normally do not say something like two patiences in either language.

Another common structure is:

mikið af þolinmæði – a lot of patience
(literally: much of patience)

Both mikla þolinmæði and mikið af þolinmæði are idiomatic. The sentence you have uses the first pattern, with mikla directly modifying the noun.