Ég set símann á náttborðið og fer í inniskóna.

Because símann is the accusative singular definite form of sími.

Here, símann is the direct object of set: you are putting the phone somewhere.

A quick breakdown:

• sími = a phone, phone
• síma = a phone/phone in the accusative singular
• símann = the phone in the accusative singular

So after ég set you need the object form, not the basic dictionary form.

Because Icelandic usually uses:

• accusative after á when something is moving onto a place
• dative after á when something is already on a place

In this sentence, the phone is being moved onto the nightstand, so you get:

• á náttborðið = onto the nightstand

Compare:

• Ég set símann á náttborðið. = I put the phone onto the nightstand.
• Síminn er á náttborðinu. = The phone is on the nightstand.

So this is a very common Icelandic pattern: motion = accusative, location = dative with certain prepositions.

Because the two prepositions reflect different spatial ideas:

• á = on, onto a surface
• í = in, into something

A nightstand is thought of as a surface you put something onto, so á is natural.

Slippers are something you put your feet into, so Icelandic uses í. This is also the normal way Icelandic talks about putting on many kinds of clothes and shoes:

• fara í skó = put on shoes
• fara í buxur = put on trousers
• fara í kápu = put on a coat

So even if English says put on slippers, Icelandic thinks more like go into the slippers.

Because fara í is a very common Icelandic expression for put on / get into clothes, shoes, and similar things.

Literally, fara means go, so word-for-word it looks like go into the slippers, but idiomatically it means:

• put on the slippers
• slip into the slippers

This is completely normal Icelandic usage, not a strange poetic expression.

Compare:

• Ég fer í inniskóna. = I put on the slippers.
• Ég er í inniskónum. = I am wearing the slippers.

The first is the action of putting them on; the second is the state of wearing them.

Because inniskóna is the correct form after í here.

In this sentence, í shows motion into the slippers, so it takes the accusative.
The noun is also definite: the slippers.

So:

• inniskór = slippers / slipper-shoes
• inniskóna = the slippers in the accusative plural

Useful contrast:

• fer í inniskóna = puts on the slippers
• er í inniskónum = is wearing / is in the slippers

So the ending changes because of case.

The dictionary forms are:

• setja = to put, set
• fara = to go

In the sentence:

• ég set = I put
• ég fer = I go / I put on, when used with í for clothes or shoes

This can be confusing at first because the form in the sentence does not always look much like the infinitive, especially with fara → fer.

Because the subject is still the same: I.

Icelandic often leaves out the repeated subject in coordinated clauses when it is obvious:

• Ég set símann á náttborðið og fer í inniskóna.

This is the same basic idea as English:

• I put the phone on the nightstand and put on my slippers.

You could also say:

• Ég set símann á náttborðið og ég fer í inniskóna.

But that sounds more explicit or slightly more emphatic. The version without the second ég is very natural.

Yes, basically. Icelandic usually adds the definite article as an ending on the noun instead of using a separate word like English the.

In this sentence:

• símann = the phone
• náttborðið = the nightstand
• inniskóna = the slippers

The exact ending changes depending on:

• gender
• number
• case

So these are not random endings; they are different forms of the definite article attached to the noun.

It is in the present tense.

But the present tense in Icelandic, like in English, can be used in more than one way. Depending on context, this sentence could describe:

• a habitual routine
• a sequence of actions happening now
• a vivid narration of what someone does

So it might mean something like:

• I put the phone on the nightstand and put on the slippers
• I’m putting the phone on the nightstand and putting on the slippers

The exact time sense depends on context, not just the verb form.

The basic order here is:

• Ég = subject
• set = verb
• símann = object
• á náttborðið = prepositional phrase
• og fer í inniskóna = second coordinated verb phrase

So the structure is basically:

Subject + verb + object + place + and + verb phrase

This is very normal Icelandic word order in a simple main clause.

It is also useful to know that Icelandic is a verb-second language in main clauses. That means if something else comes first, the verb still tends to stay in second position:

• Á kvöldin set ég símann á náttborðið...
  = In the evenings, I put the phone on the nightstand...

That is a broader grammar point, but this sentence fits the normal pattern very well.