Ég þakka henni fyrir skilninginn þegar ég þarf að breyta áætluninni á síðustu stundu.

Because þakka takes a dative object in Icelandic.

• hún = she in the nominative
• henni = her in the dative

So Ég þakka henni means I thank her. This is something you simply have to learn with the verb: að þakka einhverjum = to thank someone.

Because fyrir here takes the accusative, and skilninginn is the accusative singular definite form of skilningur.

Breakdown:

• skilningur = understanding
• skilninginn = the understanding

So:

• fyrir skilning = for understanding
• fyrir skilninginn = for the understanding

In English we often just say for understanding, but Icelandic often uses the definite form in polite expressions like this.

Because breyta takes the dative when it means change something.

So:

• að breyta einhverju = to change something

That gives:

• áætlunin = the plan
• áætluninni = the plan, in the dative

This is a very common thing learners need to memorize: some Icelandic verbs require a specific case for their object, and breyta is one of them.

Þarf is from að þurfa, meaning to need / have to.

The pattern is:

• þurfa að + infinitive

So:

• ég þarf að breyta = I need to change / I have to change

Here:

• þarf = present tense, 1st/3rd person singular form
• að breyta = to change

This is very similar to English need to change.

Because Icelandic, like English, often uses the present tense for things that happen generally, repeatedly, or whenever the situation comes up.

So þegar ég þarf að breyta áætluninni á síðustu stundu means something like:

• when I need to change the plan at the last minute
• or whenever I need to change the plan at the last minute

It does not have to mean only right now. It can describe a recurring situation.

It can feel like either, depending on context.

In this sentence, þegar introduces a time clause:

• þegar ég þarf að breyta áætluninni = when I need to change the plan

Because the main clause is general and present-tense, English may naturally understand it as whenever. So the Icelandic þegar here can cover the idea of:

• when
• whenever, in a general sense

The exact nuance comes from the whole sentence, not from þegar alone.

Here á takes the dative, and both words show that.

Breakdown:

• stunda is not the form here; the noun is stund
• stundu = dative singular of stund
• síðustu = the matching feminine singular dative form of síðastur

So:

• á síðustu stundu = at the last minute

This is a fixed and very common expression. It is best learned as a whole phrase.

The preposition á can take either accusative or dative, depending on meaning.

A useful basic rule is:

• accusative: movement toward / onto something
• dative: location, position, or a set expression

In á síðustu stundu, there is no movement. It is a time expression meaning at the last minute, so the phrase uses the dative.

This same preposition behavior happens in many places in Icelandic, so it is worth getting used to.

Icelandic uses the definite article as a suffix attached to the noun.

So:

• skilningur = understanding
• skilningurinn = the understanding
• áætlun = plan
• áætlunin = the plan

But because of case changes, in the sentence you get:

• skilninginn = accusative singular definite
• áætluninni = dative singular definite

In polite or specific contexts, Icelandic often sounds natural with the definite form where English may or may not use the.

Yes. Icelandic word order is somewhat flexible, although not completely free.

The sentence as given is natural:

• Ég þakka henni fyrir skilninginn þegar ég þarf að breyta áætluninni á síðustu stundu.

You could also move the time clause to the front:

• Þegar ég þarf að breyta áætluninni á síðustu stundu, þakka ég henni fyrir skilninginn.

Notice that when the subordinate clause comes first, the finite verb in the main clause comes before the subject:

• þakka ég
• not ég þakka

That is part of Icelandic's verb-second pattern.

Yes, very much so.

Þakka einhverjum fyrir skilninginn is a common and natural way to say that you appreciate someone's understanding, patience, or flexibility.

It works a lot like polite English phrases such as:

• I thank her for her understanding
• I appreciate her understanding

So while the grammar is fully regular, the whole expression is also something you should learn as a useful chunk.

Grammatically, henni tells you only that the person is:

• singular
• female
• in the dative

So yes, it refers to one woman / one female person. But the sentence itself does not tell you who she is. That would come from context.

If it were a man, you would use honum instead:

• Ég þakka honum... = I thank him...

A very useful set of things:

• að þakka einhverjum = to thank someone
• fyrir + accusative = for
• að breyta einhverju = to change something
• þurfa að + infinitive = need to do something
• á síðustu stundu = at the last minute

If you remember those chunks, the whole sentence becomes much easier to build and understand.