Ég brosi alltaf þegar amma kveður mig við hurðina.

Because brosa is the infinitive: að brosa = to smile.

In the sentence, the verb has to match the subject ég (I), so Icelandic uses the 1st person singular present form:

• að brosa = to smile
• ég brosi = I smile
• þú brosir = you smile
• hann/hún brosir = he/she smiles

This is one of the big differences from English: Icelandic verbs change form more often depending on the subject.

That is the most natural neutral word order here.

In a simple main clause, Icelandic often places the finite verb early, and adverbs like alltaf (always) commonly come after it:

• Ég brosi alltaf ... = I always smile ...

So this order sounds natural and standard.

You can move things around for emphasis, but the basic learner-friendly pattern here is:

• subject + finite verb + adverb

Here þegar means when and introduces a subordinate clause:

• þegar amma kveður mig við hurðina
  = when grandma says goodbye to me at the door

Be careful: þegar can also mean already in other sentences.

For example:

• Ég er þegar kominn. = I have already arrived.

So the meaning depends on how it is used:

• þegar + clause = when
• þegar as an adverb = already

Because Icelandic often leaves out the possessive with close family members when it is already clear from context.

So:

• amma can naturally mean grandma / my grandma
• amma mín = my grandma

Both are possible, but amma by itself is very common when the speaker is clearly talking about their own grandmother.

Using amma mín can sound a bit more explicit, personal, or emphatic, but it is not required.

Because kveðja is the infinitive (to greet / to say goodbye to), but the sentence needs a conjugated form.

The subject of that clause is amma (grandma), which is 3rd person singular, so the verb becomes kveður:

• að kveðja = to greet / to say goodbye to
• ég kveð = I greet / I say goodbye
• amma kveður = grandma greets / grandma says goodbye

This verb is not conjugated in a completely obvious way, so it is one you mainly learn as a pattern.

In this sentence, kveður mig means says goodbye to me.

The verb að kveðja can have a few related meanings depending on context, such as:

• to greet
• to bid farewell to
• to say goodbye to

Here, because the action happens við hurðina (at the door), the farewell meaning is the natural one.

So:

• amma kveður mig við hurðina
  = grandma says goodbye to me at the door

Because mig is the accusative form of ég.

The verb að kveðja takes a direct object, and that object is put in the accusative. So:

• ég = I (subject form, nominative)
• mig = me (object form, accusative)
• mér = to me / for me (dative in many uses)

Here grandma is doing the action, and me is the person receiving it, so Icelandic uses:

• amma kveður mig

not ég, because ég can only be the subject.

Because Icelandic expresses this idea differently from English.

In English, we say:

• say goodbye to me

But Icelandic uses the verb að kveðja directly with an object:

• kveðja mig

So the to is built into how the Icelandic verb works. You do not need an extra preposition there.

A useful way to think about it is:

• English: say goodbye to someone
• Icelandic: kveðja einhvern

við hurðina means at the door or by the door.

Here is the breakdown:

• við = by / at
• hurðina = the door

The noun comes from hurð (door), and -ina is the ending for the definite accusative singular form here:

• hurð = door
• hurðin = the door
• hurðina = the door (accusative form)

The preposition við takes the accusative, so you get við hurðina.

Yes. You can put the when clause first:

• Þegar amma kveður mig við hurðina, brosi ég alltaf.

That is perfectly natural.

But notice what happens in the main clause:

• brosi ég alltaf
• not ég brosi alltaf

This is because Icelandic main clauses follow the verb-second pattern very often. When the subordinate clause comes first, it takes the first position, so the finite verb of the main clause comes next.

So compare:

• Ég brosi alltaf þegar amma kveður mig við hurðina.
• Þegar amma kveður mig við hurðina, brosi ég alltaf.

Both mean the same thing, but the word order changes.

It normally expresses a habitual action: something that happens regularly.

That comes from:

• the present tense
• the adverb alltaf (always)

So the sense is:

• Whenever grandma says goodbye to me at the door, I smile.

It does not usually sound like one single event. It sounds like something that happens again and again.