Hvolpurinn gengur í taumi.

Because Icelandic usually adds the definite article to the end of the noun instead of using a separate word like English the.

• hvolpur = puppy
• hvolpurinn = the puppy

So the -inn here is the definite article attached to a masculine singular noun in the nominative.

You would just say hvolpur.

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

• hvolpur = a puppy / puppy
• hvolpurinn = the puppy

Which one is meant depends on whether the definite ending is present and on the context.

Gengur is the 3rd person singular present tense of the verb ganga.

So it matches a singular subject like hvolpurinn.

• að ganga = to walk, to go
• ég geng = I walk
• hann/hún/það gengur = he/she/it walks

In this sentence, gengur agrees with hvolpurinn, which is singular.

It can mean either, depending on context.

Icelandic often uses the simple present where English might use either:

• walks
• is walking

So Hvolpurinn gengur í taumi can describe a general situation or something happening now, depending on the context around it.

Because prepositions do not match one-to-one between languages.

The preposition í often means in, but in some expressions it corresponds to something different in English. Í taumi is an idiomatic way to say on a leash.

So this is a good phrase to learn as a whole:

• ganga í taumi = to walk on a leash

It is not a good idea to translate í mechanically as in every time.

Because taumi is the dative singular form of the noun taumur.

The noun’s basic dictionary form is:

• taumur = leash, rein

But after í, Icelandic often uses the dative when describing a state, position, or condition rather than motion into something. In this expression, that gives:

• í taumi

So the change from taumur to taumi is a case change.

No. Í can take either accusative or dative.

A very common rule is:

• accusative = movement into
• dative = location/state

In í taumi, the idea is not that the puppy is moving into a leash. It is describing the condition the puppy is in: it is walking while being on a leash. That is why the dative is used here.

The basic order here is:

• Subject: Hvolpurinn
• Verb: gengur
• Prepositional phrase: í taumi

So it is a normal, straightforward Icelandic sentence.

However, Icelandic word order is somewhat flexible, especially because of the verb-second tendency in main clauses. In neutral style, though, Hvolpurinn gengur í taumi is the most natural order.

Yes. A simple yes/no question would be:

• Gengur hvolpurinn í taumi?

That puts the verb first, which is very common in Icelandic questions.

So the difference is:

• Hvolpurinn gengur í taumi. = statement
• Gengur hvolpurinn í taumi? = question