Hún skilur þágufall betur eftir góða útskýringu.

No. This is a very common point of confusion.

The word þágufall means dative case, but inside this sentence it is just the object of the verb skilur. The verb skilja normally takes an accusative object.

So in this sentence, þágufall is functioning as an accusative object, not as a dative noun. It just happens to look the same because þágufall is a neuter noun, and the nominative and accusative singular forms are identical.

A useful comparison:

• nominative: þágufall
• accusative: þágufall
• dative: þágufalli

Icelandic often leaves the noun indefinite when talking about a subject or category in general, especially with technical or school-related terms.

So skilur þágufall is natural for understands the dative / dative case.

If you wanted a definite form, Icelandic usually adds it to the end of the noun:

• þágufallið = the dative case

Skilja is the infinitive, meaning to understand.
Skilur is the 3rd person singular present tense, which matches hún (she).

So:

• að skilja = to understand
• ég skil = I understand
• þú skilur = you understand
• hún skilur = she understands

Because betur is an adverb, and it modifies the verb skilur.

Here the meaning is understands better, so Icelandic uses the adverb betur.

Compare:

• Hún skilur betur. = She understands better.
• Þetta er betra svar. = This is a better answer.

In the second example, betra is an adjective describing the noun svar.

Also, this form is irregular:

• vel = well
• betur = better

Because eftir here means after in a temporal/result sense, and in this use it takes the accusative.

The noun útskýring is feminine, and its accusative singular form is útskýringu. The adjective has to agree with it, so you get:

• góða útskýringu = accusative singular feminine

A useful contrast:

• góða útskýringu = accusative
• góðri útskýringu = dative

The adjective tells you.

The noun form útskýringu can appear in both accusative and dative singular, but the adjective changes:

• góða útskýringu = accusative
• góðri útskýringu = dative

So even if the noun alone is ambiguous, the full phrase is not.

Not in every use. As with many Icelandic prepositions, you have to learn the case together with the meaning.

For this sentence, the important point is:

• eftir + accusative when it means after in time/result

So eftir góða útskýringu is the pattern to remember here.

That word order is natural and neutral in Icelandic:

• Hún = subject
• skilur = verb
• þágufall = object
• betur = adverb
• eftir góða útskýringu = prepositional phrase

Icelandic word order is somewhat flexible, but this version sounds straightforward and normal. The sentence is basically built as:

She understands + the dative better + after a good explanation.

Hún is in the nominative because it is the subject of the sentence.

Compare:

• hún = she (subject form)
• hana = her (object form)

So Hún skilur... means She understands...

Þ is pronounced like the th in thing, not like the th in this.

So the beginning of þágufall starts with that voiceless th sound:

• þ = English th in thing

A good extra contrast is:

• þ = thing
• ð = more like this