Hún notar garðhanska þegar hún tekur illgresi og heldur á skóflu.

Because þegar introduces a new clause, and Icelandic normally states the subject again in that new clause:

• Hún notar garðhanska
• þegar hún tekur illgresi og heldur á skóflu

Even though it is the same person, you usually cannot just leave out hún there.

Because tekur and heldur are two coordinated verbs with the same subject inside the same clause.

So þegar hún tekur illgresi og heldur á skóflu means:

• she pulls weeds
• and she holds a shovel

Both actions belong to the same hún, so repeating the subject is unnecessary unless you want extra emphasis.

Þegar means when and introduces a time clause.

Here it tells you when she uses the gardening gloves:

• Hún notar garðhanska þegar ... = She uses gardening gloves when ...

So the second part gives the situation in which the first action happens.

After a subordinating word like þegar, Icelandic normally uses clause order with the subject before the verb:

• þegar hún tekur ...

The order þegar tekur hún ... would sound like main-clause or question-style word order, not the normal subordinate-clause pattern.

Because nota takes an object, and that object is in the accusative case. Here garðhanska is the accusative form.

This noun is a good example of how Icelandic endings can be a little tricky:

• dictionary form: garðhanski
• accusative form used here: garðhanska

In context, it means gardening gloves. The form can look the same as singular in some patterns, so context helps you understand the number.

Icelandic often uses a bare noun when the meaning is indefinite or general.

So:

• notar garðhanska = uses gardening gloves
• tekur illgresi = pulls weeds
• heldur á skóflu = holds a shovel

If the noun were definite, Icelandic would usually attach the article to the end of the noun instead:

• skófluna = the shovel

Literally, taka usually means take, but in this expression taka illgresi means pull weeds, remove weeds, or do weeding.

So this is a natural Icelandic way to talk about getting weeds out of the ground, not about physically taking possession of them.

Illgresi is often used as a collective noun, so a singular-looking form can refer to weeds in general.

That means Icelandic does not always need a clearly plural form where English uses weeds. In this sentence, illgresi means weed growth / weeds as a general category.

Because halda á is the usual expression for physically holding something in your hands.

So:

• halda á einhverju = hold/carry something in the hand(s)

Without á, halda often has other meanings, such as keep, maintain, continue, or similar ideas. So the preposition is important here.

Because skófla changes form after á in the expression halda á.

The noun is in the dative case here:

• nominative: skófla
• dative: skóflu

So heldur á skóflu is the grammatically correct form for holds a shovel.

It can very naturally describe a habitual action.

So this sentence can mean something like:

• she uses gardening gloves when she pulls weeds and holds a shovel

That is a general, repeated situation, not necessarily one single event happening at this exact moment.

Yes. Garðhanski is a compound:

• garð- = garden
• hanski = glove

A very common feature of Icelandic is that compounds are built by joining nouns or stems together. The final part of the compound usually carries the grammatical ending, which is why you see garðhanska here.