Ég nota svuntu þegar ég elda, en hún notar viskustykki þegar hún tekur heitan pott af eldavélinni.

Because the verb að nota changes by person.

In the present tense:

• ég nota = I use
• þú notar = you use
• hún/hann/það notar = she/he/it uses

So -a is the usual 1st person singular ending, and -ar is the usual 2nd/3rd person singular ending for this kind of verb.

That is a very common point of confusion.

The verb elda has:

• infinitive: að elda = to cook
• 1st person singular present: ég elda = I cook / I am cooking

So the form elda can look the same in both cases. The difference is the context:

• að elda = infinitive
• ég elda = finite verb, present tense

In þegar ég elda, the presence of ég shows that this is a full clause, so elda means I cook, not to cook.

Because svunta is the direct object of nota, and nota takes the accusative case.

The dictionary form is:

• svunta = apron

But in the accusative singular it becomes:

• svuntu

So:

• Ég nota svuntu = I use an apron

This is a normal pattern for many feminine nouns ending in -a.

Because viskustykki is a neuter noun, and in the singular, many neuter nouns have the same form in the nominative and accusative.

So:

• nominative: viskustykki
• accusative: viskustykki

That is why it does not visibly change, even though it is also a direct object after notar.

It comes from the verb að taka = to take.

Its present-tense forms are irregular:

• ég tek = I take
• þú tekur = you take
• hún tekur = she takes

So tekur is the correct 3rd person singular present form.

It is not takar because að taka is not a regular -a verb in the present tense.

Because both the noun and the adjective have to match the grammar of the sentence.

The dictionary forms are:

• pottur = pot
• heitur = hot

But here pottur is the direct object of tekur, so it must be accusative singular:

• pottur → pott

And the adjective must agree with it in gender, number, and case:

• masculine
• singular
• accusative

So:

• heitur pottur = a hot pot (nominative)
• heitan pott = a hot pot (accusative)

Because af means from/off, and in this meaning it takes the dative case.

The noun is:

• eldavél = stove

With the definite article:

• eldavélin = the stove

After af, it must be dative:

• af eldavél
• af eldavélinni = off/from the stove

So the form changes because the preposition af requires it.

It shows two things at once:

1. definiteness = the
2. dative singular feminine

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

So:

• eldavél = stove
• eldavélin = the stove
• eldavélinni = the stove, in the dative form

In this sentence, af requires the dative, so you get eldavélinni.

Because þegar introduces a subordinate clause.

In Icelandic, subordinate clauses usually have normal subject + verb order:

• þegar ég elda
• þegar hún tekur

So this is different from the main-clause verb-second pattern that Icelandic often has.

A learner can safely remember:

• after þegar, expect a normal subordinate clause

Because en joins two separate clauses, and the second clause needs its own subject.

Here the sentence has:

• Ég nota svuntu þegar ég elda
• en hún notar viskustykki þegar hún tekur heitan pott af eldavélinni

Since the subject changes from ég to hún, Icelandic must state hún explicitly.

Because af means off/from a surface or position on something, while úr means out of/from inside something.

A pot sits on the stove, so you take it:

• af eldavélinni = off the stove

You would use úr for something coming out of the inside of something:

• úr pottinum = out of the pot

So af is exactly the right choice here.

In this sentence, yes, þegar means when.

That said, Icelandic þegar can have a few related uses depending on context, including when and sometimes as soon as. But in a sentence like this, the natural understanding is simply when.

So a good beginner reading is:

• þegar ég elda = when I cook
• þegar hún tekur... = when she takes...