Ég veit ekki leiðina heim.

Word by word:

• Ég = I
• veit = know (present tense of vita)
• ekki = not
• leiðina = the way / the route
• heim = home, or more literally homewards

So the whole sentence means I do not know the way home or I don't know the way home.

Because leiðina is the definite accusative singular form of leið.

The noun leið means way, route, path.
In this sentence, Icelandic uses the equivalent of the way, not just way, so it becomes leiðin in the nominative definite form, and leiðina in the accusative definite form.

A rough breakdown is:

• leið = way
• leiðin = the way
• leiðina = the way (as an object)

Here it is the direct object of veit, so the accusative form is used.

Because it functions as the direct object of the verb.

In Icelandic, many verbs require their object to be in a particular case.
With vita (to know), the thing known is commonly in the accusative.

So:

• Ég veit = I know
• Ég veit leiðina = I know the way

Since leiðina is the thing being known, it appears in the accusative.

This is normal Icelandic word order.

In a simple main clause, Icelandic usually puts the finite verb in the second position. This is often called the V2 rule.

So the structure here is:

• Ég = first element
• veit = finite verb in second position
• ekki = negation after the verb

That is why Icelandic says:

• Ég veit ekki ...

rather than something like Ég ekki veit ...

This is one of the big differences from English.

Heim means home in a directional sense: to home, homeward, or home as a destination.

That is why Icelandic does not need a preposition here. English says the way home, and Icelandic does the same kind of thing with heim.

Compare:

• heima = at home
• heim = home / to home

So:

• Ég er heima = I am at home
• Ég fer heim = I am going home

In your sentence, leiðina heim means the way home.

Because heim itself already expresses movement or direction toward home.

English also often omits a preposition in go home or the way home. Icelandic works similarly here.

So Icelandic says:

• leiðina heim = the way home

not something like leiðina til heim.

If you want to say to the house/home in a different sense, other constructions are possible, but heim by itself is the normal word for homeward / home.

Usually, veit is the natural choice here.

• vita = to know a fact, piece of information, answer, route, etc.
• kunna can sometimes mean know in the sense of know how, be familiar with, or have learned

With leiðina, Icelandic very commonly uses vita:

• Ég veit ekki leiðina heim = I don't know the way home

You may also hear kunna leiðina in some contexts, especially with the meaning know the route / be familiar with the route, but for a learner, vita is the safest and most straightforward match here.

It could be, depending on nuance.

Icelandic can express the way home in more than one way:

• leiðina heim = literally the way home
• heimleiðina = the homeward route / the way home

The sentence you have is very natural and transparent for learners. It uses a noun phrase plus the directional adverb heim.

So leiðina heim is not strange at all—it is a normal, clear way to say it.

Yes, but the finite verb still normally stays in second position in a main clause.

For example, you could front another element for emphasis:

• Leiðina heim veit ég ekki. = The way home, I don't know.
• Heim veit ég ekki leiðina. = possible, but much less neutral

The most neutral everyday order is:

• Ég veit ekki leiðina heim.

So yes, word order can change, but the original sentence is the standard, unmarked version.

A yes/no question is formed by putting the verb first:

• Veist þú ekki leiðina heim? = Don't you know the way home?
• Veistu leiðina heim? = Do you know the way home?

A few useful notes:

• veist is the you form of vita
• þú = you
• In everyday Icelandic, veist þú is often written/spoken as veistu

You would use the past tense of vita:

• Ég vissi ekki leiðina heim. = I didn't know the way home.

Here:

• vissi = knew

So the pattern is:

• Ég veit ekki ... = I don't know ...
• Ég vissi ekki ... = I didn't know ...

Yes, it is natural and idiomatic.

A native speaker would understand it immediately as I don't know the way home.

Depending on context, other natural alternatives might also be used, such as:

• Ég veit ekki hvernig ég á að komast heim. = I don't know how to get home.
• Ég kann ekki leiðina heim. = I don't know the route home / I'm not familiar with the way home.

But your original sentence is perfectly good Icelandic.