Launamiði framteljandans er réttur.

That is the normal Icelandic way to show possession with a noun: [thing] + [owner in the genitive].

So launamiði framteljandans is literally something like pay slip of the taxpayer. English usually prefers the taxpayer's pay slip, but Icelandic often puts the possessed thing first and the possessor after it.

It shows that framteljandans is genitive singular definite.

The dictionary form is framteljandi.

• framteljandi = a taxpayer / declarant
• framteljanda = of a taxpayer
• framteljandans = of the taxpayer

So framteljandans means the taxpayer's or of the taxpayer.

Launamiði is nominative singular, because it is the subject of the sentence.

Framteljandans is genitive singular, because it expresses possession.

So the structure is:

• launamiði = subject
• framteljandans = genitive modifier, showing whose pay slip it is

Because réttur has to agree with launamiði.

Launamiði is:

• masculine
• singular
• nominative

So the adjective also appears in the masculine singular nominative form: réttur.

If the subject were neuter or feminine, the form would be different.

After er (is), Icelandic predicate adjectives normally agree with the subject and are usually in the nominative.

So in Launamiði framteljandans er réttur, both the subject noun and the predicate adjective are nominative:

• launamiði
• réttur

This is very common with vera (to be).

Icelandic usually does not use a separate word like English the. Instead, definiteness is often shown with an ending attached to the noun.

In a phrase like launamiði framteljandans, Icelandic can leave the first noun without a visible definite ending, even though English naturally translates the whole phrase as definite: the taxpayer's pay slip.

If you wanted a more explicitly definite head noun, you could also see launamiðinn framteljandans.

Yes. It is made from:

• laun = wages, pay
• miði = slip, ticket, note

So launamiði is literally something like pay-slip or wage-slip. Icelandic makes a lot of compound nouns this way, so recognizing the parts is very useful.

Because Icelandic main clauses usually follow a verb-second pattern.

That means the finite verb often comes in the second slot of the clause, not necessarily as the second single word. Here the first slot is the whole noun phrase Launamiði framteljandans, and then the verb er comes next.

So the structure is:

• first element: Launamiði framteljandans
• second element: er
• rest: réttur

They are:

• launamiði — pay slip / wage slip
• framteljandi — taxpayer, declarant, person filing a return
• vera — to be
• réttur — correct, right

This is useful because Icelandic words often appear in inflected forms, so the form you see in the sentence is not always the dictionary form.

Yes, depending on what you want to emphasize.

Some possible alternatives are:

• Launamiðinn er réttur.
  The pay slip is correct.

• Launamiði hans er réttur.
  His pay slip is correct.

• Launamiðinn framteljandans er réttur.
  A more explicitly definite version of the original structure.

But launamiði framteljandans is a normal and natural way to say the taxpayer's pay slip.