Á þessum runna eru ný lauf og nýjar greinar.

Icelandic often puts the location first when that is the setting or the thing being highlighted.

So Á þessum runna eru... is a natural way to say something like On this bush, there are...

A more neutral subject-first version would be:

• Ný lauf og nýjar greinar eru á þessum runna.

Both are correct, but the given sentence gives extra focus to on this bush.

Because Icelandic main clauses usually follow the verb-second rule.

That means:

• one element comes first
• then the finite verb comes second

Here, the first element is Á þessum runna, so the verb eru must come next:

• Á þessum runna
  • eru
    • ný lauf og nýjar greinar

This word order is very common in Icelandic.

Because á means on or in/on depending on context, and here it introduces the place where something is located.

So:

• á þessum runna = on this bush

It is a prepositional phrase telling you where the leaves and branches are.

Because þessi changes form depending on gender, number, and case.

The noun runni is:

• masculine
• singular

And after á in a location meaning, it is in the dative.

So the correct form is:

• þessum runna

Compare:

• þessi runni = this bush (nominative)
• á þessum runna = on this bush (dative)

The dictionary form is runni = bush / shrub.

But this noun changes form by case. Here it is in the dative singular, so runni becomes runna.

So:

• nominative: runni
• dative: runna

That is why the sentence has á þessum runna.

Because with the preposition á, Icelandic normally uses:

• dative for location
• accusative for motion onto

This sentence describes where the leaves and branches are, not movement. So the dative is used:

• á þessum runna = on this bush

Compare:

• á þessum runna = on this bush
• á þennan runna = onto this bush

Because adjectives in Icelandic must agree with the nouns they describe.

Here:

• lauf is neuter plural
• greinar is feminine plural

So nýr has to appear in different forms:

• ný with neuter plural: ný lauf
• nýjar with feminine plural: nýjar greinar

English uses just new for both, but Icelandic changes the adjective form.

Because the two nouns have different grammatical genders, so the adjective needs a different form for each one.

• ný lauf
• nýjar greinar

You cannot use one single adjective form to agree with both at the same time. Repeating the adjective is the normal solution.

They are in the nominative, because they are the subject of the verb eru.

So:

• lauf = nominative plural neuter
• greinar = nominative plural feminine

Even though the sentence begins with the location phrase, the things that are on the bush are still the subject.

Here it is understood as plural: leaves.

A useful thing to know is that lauf can look the same in more than one form, so context matters. In this sentence, the adjective helps:

• ný lauf = new leaves

The adjective form ný is the neuter plural form here.

English often says:

• There are new leaves and new branches on this bush.

But Icelandic usually does not need a dummy word like there in this kind of sentence. It simply uses the verb vera:

• eru ný lauf og nýjar greinar

So the Icelandic structure is closer to:

• On this bush are new leaves and new branches.

That sounds unusual in English, but it is normal in Icelandic.

Because Icelandic does not use an indefinite article like English a/an.

So bare nouns can mean things like:

• lauf = leaves
• greinar = branches

If Icelandic wants to be definite, it often uses the suffixed article instead. But here the nouns are indefinite, so no article appears.

Yes. That is fully grammatical.

The difference is mainly one of focus:

• Á þessum runna eru ný lauf og nýjar greinar.
  Focuses first on the location.

• Ný lauf og nýjar greinar eru á þessum runna.
  Focuses first on the leaves and branches.

Both are good Icelandic.