Á leiðinni heim finnum við annan stíg sem liggur við lítinn dal.

This is a normal Icelandic word order pattern. Icelandic is a verb-second language, which means the finite verb often comes in the second position of the clause.

So in:

Á leiðinni heim finnum við annan stíg ...

the first element is the adverbial phrase Á leiðinni heim = on the way home, and then the verb finnum comes second. The subject við follows the verb.

A more neutral order would be:

Við finnum annan stíg á leiðinni heim ...

but the original version puts more focus on on the way home.

Literally, it means something like on the way home.

Breakdown:

• á = on
• leiðinni = the way / the route in the dative singular
• heim = homeward / home

So the phrase is not translated word-for-word into natural English, but its sense is:

while on the way home or on the way home.

Because the preposition á here takes the dative, and the noun is also in the definite form.

Base noun:

• leið = way, route

Definite singular nominative:

• leiðin = the way

Dative singular definite:

• leiðinni = the way after a dative-governing preposition

So:

• á leið would not be correct here
• á leiðinni = on the way

This phrase is very common in Icelandic.

Because heim is an adverb meaning home(wards), especially with motion toward home.

Compare:

• heim = home, homewards (motion toward home)
• heima = at home (location)

So:

• fara heim = go home
• vera heima = be at home

In your sentence, heim works naturally because the idea is movement: on the way home.

Because finna takes a direct object in the accusative case, and stígur is a masculine noun.

Base noun:

• stígur = path, trail

Accusative singular:

• stíg

The adjective/pronoun annar also has to match the noun in gender, number, and case:

• masculine
• singular
• accusative

So:

• annar stígur = another path (nominative)
• annan stíg = another path (accusative, as the object of finnum)

The dictionary form is annar, meaning other or another.

In the sentence, it appears as annan because it has been declined to match stíg:

• masculine
• singular
• accusative

So:

• annar = dictionary form
• annan = accusative masculine singular

This is a very common kind of agreement in Icelandic.

Because stígur is in the accusative singular, not the nominative.

Many masculine nouns in Icelandic have -ur in the nominative singular, but that ending often changes or disappears in other cases.

For stígur:

• nominative: stígur
• accusative: stíg
• dative: stíg
• genitive: stígs

Since stíg is the direct object of finnum, the accusative is used.

This is a relative clause describing annan stíg.

• sem = that / which
• liggur = lies / runs / goes
• við lítinn dal = by a small valley

So the whole part means:

that runs by a small valley
or
which lies next to a small valley

It tells you more about the path.

No, sem is very convenient because it does not change form.

Unlike some relative pronouns in other languages, Icelandic sem stays the same whether it refers to a masculine, feminine, neuter, singular, or plural noun.

So in this sentence:

annan stíg sem liggur ...

sem simply means that/which and introduces the relative clause.

Liggur is the present tense of the verb liggja, which often means:

• lie
• be situated
• run (for roads, paths, rivers, etc.)

In this sentence, the path is the subject of the relative clause:

stígur sem liggur við lítinn dal
= a path that lies / runs by a small valley

So liggur agrees with the singular subject stígur.

Because the preposition við governs the accusative in this usage.

Base noun:

• dalur = valley

Accusative singular:

• dal

The adjective lítill must match it:

• masculine
• singular
• accusative

So:

• lítill dalur = a small valley (nominative)
• lítinn dal = a small valley (accusative)

That is why the phrase is:

við lítinn dal

Because lítill (small) is an irregular adjective.

Its forms do not follow the most regular adjective pattern. For example:

Masculine singular:

• nominative: lítill
• accusative: lítinn
• dative: litlum
• genitive: lítils

So lítinn dal is just the correct accusative masculine singular form.

This is a good adjective to memorize early because it is very common.

Icelandic does not have an independent indefinite article like English a/an.

So:

• stígur can mean path or a path
• dalur can mean valley or a valley

Indefiniteness is usually understood from context.

That is why:

• annan stíg = another path
• lítinn dal = a small valley

There is no separate word corresponding to English a.

Yes, very much so.

Á leiðinni is a very common expression meaning:

• on the way
• en route

You will often see it with another destination word:

• á leiðinni heim = on the way home
• á leiðinni í skólann = on the way to school
• á leiðinni til Akureyrar = on the way to Akureyri

So it is worth learning as a chunk, not just as separate words.

No, not in this sentence.

Use:

• heim for motion toward home
• heima for being at home

So:

• á leiðinni heim = on the way home
• heima would suggest location, not direction, and would sound wrong here

This is a very common contrast in Icelandic.

A useful way to see it is:

Á leiðinni heim = adverbial phrase
finnum við = verb + subject
annan stíg = direct object
sem liggur við lítinn dal = relative clause describing the object

So the structure is:

[On the way home] [find we] [another path] [that runs by a small valley].

That helps explain both the word order and the case forms.