Brekkan upp að húsinu er hál þegar það rignir.

The -n is the Icelandic definite article, attached to the noun. So:

• brekka = slope, hill
• brekkan = the slope

Here brekkan is also in the nominative singular, because it is the subject of the sentence.

It means up to the house or leading up to the house.

• upp = up
• að = to, toward, up to
• húsinu = the house in the form required after að

Together, upp að húsinu describes the slope as the one that goes upward toward the house.

Because að takes the dative case.

The noun hús is neuter:

• húsið = the house in nominative/accusative
• húsinu = the house in dative

Since the phrase is að húsinu, the dative is required.

Because Icelandic adjectives agree with the noun they describe.

The subject is brekkan, which is:

• feminine
• singular
• nominative

So the adjective must also be feminine singular nominative:

• masculine: háll
• feminine: hál
• neuter: hált

That is why the sentence has Brekkan ... er hál.

It is a dummy subject, just like English it in it rains.

It does not refer to any actual thing. It is simply there because Icelandic weather expressions often use það:

• Það rignir = It rains
• Það snjóar = It snows

So in this sentence, þegar það rignir means when it rains.

Because the sentence is expressing a general truth or habitual situation, not one specific rainy moment.

So þegar það rignir here is best understood as:

• when it rains
• whenever it rains

Icelandic often uses the present tense this way, just like English does in sentences such as Ice melts when it gets warm.

Þegar can indeed mean two different things:

• already as an adverb
• when as a conjunction

In this sentence, it is followed by a whole clause, það rignir, so it is clearly the conjunction when:

• þegar það rignir = when it rains

If it meant already, it would be functioning differently in the sentence.

Because the speaker is talking about a specific slope and a specific house.

So:

• brekkan = the slope
• húsinu = the house

This is very natural in Icelandic. The sentence suggests a known place: the slope leading up to that particular house.

The sentence breaks down like this:

• Brekkan upp að húsinu = the subject phrase
• er = the verb is
• hál = predicate adjective
• þegar það rignir = time clause, when it rains

So the overall structure is:

[subject] + [verb] + [adjective] + [subordinate clause]

Yes. You can say:

Þegar það rignir er brekkan upp að húsinu hál.

That means the same thing.

This is a good example of Icelandic word order: in a main clause, the finite verb usually stays in the second position. So when the þegar-clause comes first, er still comes next.