Hún er nýbúin að gróðursetja fræ við hliðina á runnanum.

This is a very common Icelandic way to say that someone has just finished doing something or has just done something.

So Hún er nýbúin að gróðursetja ... means something like:

• She has just planted ...
• She has just finished planting ...

The key part is nýbúin að:

• búin að = finished doing / done with
• nýbúin að = just finished doing / has just done

It is not a literal one-word equivalent of the English perfect, but it often translates that way.

Because it agrees with hún, which is feminine singular.

The dictionary form is nýbúinn, but adjectives in Icelandic change for gender, number, and case. Here the subject is hún = she, so the form is feminine singular nominative:

• hann er nýbúinn = he has just finished
• hún er nýbúin = she has just finished
• þau eru nýbúin = they have just finished

So nýbúin is simply the correct agreement form for a female subject.

Here að is the infinitive marker, basically like English to before a verb.

So:

• að gróðursetja = to plant

After vera búinn / búin að, Icelandic normally uses að + infinitive:

• Ég er búinn að borða. = I have finished eating.
• Hún er nýbúin að lesa bókina. = She has just finished reading the book.

So in your sentence, að introduces the action that she has just completed.

Yes. Gróðursetja means to plant, especially in the sense of putting something into the ground so it can grow.

It is a good fit for things like:

• seeds
• flowers
• trees
• shrubs

You may also see planta, which can also mean to plant, but gróðursetja is a very standard and clear word in this context.

So gróðursetja fræ means to plant seeds.

It can be tricky, because fræ has the same form in nominative/accusative singular and plural.

So fræ can mean:

• a seed
• seeds

In this sentence, context and translation tell you it means seeds.

That is normal in Icelandic: sometimes number is understood from context rather than from a visibly different form.

Because fræ is indefinite here.

The sentence is talking about planting seeds in a general sense, not a specific known set of seeds. So Icelandic simply uses the bare noun:

• gróðursetja fræ = plant seeds

If you wanted the seeds, Icelandic would normally mark definiteness on the noun itself, not with a separate word like English the.

við hliðina á is a fixed expression meaning:

• next to
• beside
• at the side of

So:

• við hliðina á runnanum = next to the bush

It is best learned as a whole phrase rather than word by word.

Literally, it is built from:

• við
• hliðina
• á

But for a learner, the most useful thing is simply to remember:

• við hliðina á + dative noun = next to + noun

Examples:

• við hliðina á húsinu = next to the house
• við hliðina á bílnum = next to the car

Because runnanum is the dative singular definite form of runni (bush, shrub), and the phrase við hliðina á requires the noun after á to be in the dative.

So:

• base form: runni
• á runnanum = on/at the bush, or in this phrase, next to the bush

The ending -anum tells you it is:

• masculine
• singular
• dative
• definite

That is why you get runnanum, not runni or runnann.

Yes, but the meaning changes slightly.

For example:

• Hún gróðursetti fræ við hliðina á runnanum.

That means She planted seeds next to the bush.

This is perfectly fine, but it does not emphasize that the action has just been completed. The original sentence does emphasize that recent completion:

• Hún er nýbúin að gróðursetja fræ við hliðina á runnanum.
  = She has just planted / has just finished planting seeds next to the bush.

So the choice depends on the nuance you want.

The main thing that changes is nýbúin, because it agrees with the subject.

For example:

• Hann er nýbúinn að gróðursetja fræ ... = He has just planted seeds ...
• Hún er nýbúin að gróðursetja fræ ... = She has just planted seeds ...
• Þær eru nýbúnar að gróðursetja fræ ... = They (feminine) have just planted seeds ...
• Þeir eru nýbúnir að gróðursetja fræ ... = They (masculine) have just planted seeds ...

The rest of the sentence can stay the same unless something else also changes.