Hann er ástfanginn.

• Hann = he
• er = is
• ástfanginn = an adjective meaning in love

So the sentence structure is:

subject + verb + predicate adjective

That is why ástfanginn is not acting like a noun here. It describes hann.

Er is the 3rd person singular present form of vera, the verb to be.

Because hann means he, you use the he/she/it is form:

• ég er = I am
• þú ert = you are
• hann er = he is

So Hann er ástfanginn literally follows the same pattern as English He is happy or He is tired.

It is an adjective.

In this sentence, it works as a predicate adjective, meaning it comes after vera (to be) and describes the subject.

So this is similar to:

• Hann er þreyttur = He is tired
• Hann er glaður = He is happy
• Hann er ástfanginn = He is in love

Because the adjective has to agree with the person it describes.

Here, it describes hann, which is masculine singular nominative, so the masculine form is ástfanginn.

For comparison:

• Hann er ástfanginn = He is in love
• Hún er ástfangin = She is in love

So the ending changes with gender and number. Also, this -inn is not the definite article here. It is just part of the adjective form.

You change the adjective so that it agrees with the subject.

Common forms are:

• Hann er ástfanginn = He is in love
• Hún er ástfangin = She is in love

For they, the adjective depends on gender:

• Þeir eru ástfangnir = They are in love (masculine or mixed group)
• Þær eru ástfangnar = They are in love (all feminine)
• Þau eru ástfangin = They are in love (neuter/mixed in certain contexts)

So yes, the form of ástfanginn changes.

You usually add af + dative.

Examples:

• Hann er ástfanginn af henni. = He is in love with her.
• Hann er ástfanginn af honum. = He is in love with him.
• Hann er ástfanginn af Maríu. = He is in love with María.

This is a very common pattern:

vera ástfanginn af einhverjum = to be in love with someone

They are related, but they are not the same.

• vera ástfanginn = to be in love
• elska einhvern = to love someone

So:

• Hann er ástfanginn af henni. = He is in love with her.
• Hann elskar hana. = He loves her.

Ástfanginn usually emphasizes the emotional state of being romantically in love.
Elska is the verb to love, and it can be romantic, but it can also be used more broadly.

Yes, it can absolutely stand by itself.

Hann er ástfanginn is a complete sentence. It means that he is in love, even if you do not say with whom.

If you want to add that information, you can:

• Hann er ástfanginn af henni.

But without that extra part, the sentence is still perfectly natural.

A rough learner-friendly approximation is:

OWST-fang-in

A few helpful points:

• á sounds roughly like ow in now
• stress is usually on the first syllable
• ng is pronounced as a single nasal sound, like in English sing

So the whole sentence is roughly:

Hann er OWST-fang-in

That said, Icelandic pronunciation has details that English spelling cannot show very well, so it is best to listen to native audio if possible.

Yes. That is very common.

• vera ástfanginn = to be in love
• verða ástfanginn = to fall in love / become in love

So:

• Hann er ástfanginn. = He is in love.
• Hann verður ástfanginn. = He falls in love / He becomes in love.

This is a useful contrast in Icelandic:

• vera describes a state
• verða describes a change into that state