Koddinn er mýkri en sófinn.

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

So:

• koddi = pillow
• koddinn = the pillow
• sófi = sofa
• sófinn = the sofa

In this sentence, both nouns are definite.

Yes. Both koddi and sófi are masculine nouns.

That matters because Icelandic nouns have grammatical gender, and gender affects:

• the form of the definite article
• adjective agreement
• pronouns
• other endings in declension

So the -inn ending here fits their masculine singular nominative form.

Er is the present tense of vera, the verb to be.

Here it simply means is:

• Koddinn er ... = The pillow is ...

Because the sentence uses the comparative form.

• mjúkur = soft
• mýkri = softer

So mýkri is the form you use when comparing two things.

Also, because it is a predicate adjective after er, it agrees with the subject. Here the subject is singular, and the correct comparative form is mýkri.

This is a stem-changing comparative.

The basic adjective is mjúkur, but in the comparative the stem changes:

• mjúkur → mýkri

So it is not just a matter of adding one simple ending. The vowel changes too. This kind of change is common in Icelandic, so it is best to learn the positive and comparative together.

Here en means than.

It introduces the second part of the comparison:

• mýkri en sófinn = softer than the sofa

Be aware that en can also mean but in other sentences, but after a comparative adjective like mýkri, it means than.

Because the comparison is understood as a shortened version of a full clause.

The sentence can be understood as:

• Koddinn er mýkri en sófinn [er].
• literally: The pillow is softer than the sofa [is].

Since sófinn is understood as the subject of that omitted part, the nominative makes sense.

Yes, but only something that is understood from context.

Icelandic, like English, often leaves out repeated words in comparisons. So the full sense is:

• The pillow is softer than the sofa is

The second er is omitted because it would be repetitive and unnecessary.

Because mýkri is being used predicatively, not directly inside the noun phrase.

In other words:

• Koddinn er mýkri = The pillow is softer
  Here the adjective comes after er.

But if the adjective directly modified the noun, it would normally come before it:

• mjúki koddinn = the soft pillow

So the position changes depending on how the adjective is being used.

It does mean the here, but the form changes depending on gender, number, and case.

So -inn is only one definite-article ending. Other forms of the same noun would use different endings.

That means you should not think of -inn as the only way to say the in Icelandic. It is just the form that fits masculine singular nominative nouns like koddinn and sófinn in this sentence.