Ég set engifer í teið þegar ég er hás.

Set is the 1st person singular present tense of the verb setja.

So:

• setja = to put, place, set
• ég set = I put

In this sentence, the most natural English translation is usually put or add, not literally set.

Because setja is the infinitive form, like to put in English.
In a normal statement, Icelandic uses a conjugated verb:

• ég set = I put
• þú setur = you put
• hann/hún/það setur = he/she/it puts

So Ég set engifer... means I put ginger...

Yes. In this sentence, I add ginger to the tea is a very natural English translation.

That is common with Icelandic verbs: the most literal English equivalent is not always the most natural one in context.

So here:

• setja í teið = literally put into the tea
• natural English = add to the tea

Here engifer is the direct object of the verb set.

A learner may expect the noun to visibly change case, but many Icelandic nouns do not show a different form in every case. In this sentence, engifer stays engifer.

Also, it is being used as a mass noun:

• engifer = ginger

So it works much like English I add ginger, without a or the.

Because the sentence is talking about ginger in general, not the ginger or a ginger as a specific countable item.

So:

• engifer = ginger
• not the ginger
• not a piece of ginger unless you say that more specifically

This is very normal with ingredients and substances.

Because í means in / into, and with this preposition Icelandic distinguishes between:

• movement into something → accusative
• location inside something → dative

Here, the ginger is being put into the tea, so Icelandic uses the accusative:

• í teið = into the tea

If you were just describing where something already is, you would use the dative instead:

• í teinu = in the tea

That is a very important Icelandic pattern.

It is the definite article attached to the noun.

So:

• te = tea
• teið = the tea

Icelandic usually puts the at the end of the noun, instead of using a separate word like English does.

Þegar means when.

It introduces the time clause:

• þegar ég er hás = when I am hoarse

So the whole sentence means something like:

• I put/add ginger into the tea when I’m hoarse.

Because Icelandic, like English, normally needs a subject in each clause.

So:

• Ég set engifer í teið = I put ginger in the tea
• þegar ég er hás = when I am hoarse

The second ég belongs to the second clause. You cannot just leave it out.

Hás means hoarse.

So:

• ég er hás = I am hoarse

It is an adjective used after the verb er (am).
That is the same basic structure as English:

• I am tired
• I am sick
• I am hoarse

Most naturally, it sounds habitual or general:

• I put ginger in the tea when I’m hoarse.
• in other words, that’s what I do whenever this happens

The present tense in Icelandic is often used this way, just like in English.

Yes. If you begin with the subordinate clause, Icelandic keeps the finite verb in second position in the main clause.

So:

• Ég set engifer í teið þegar ég er hás.
• Þegar ég er hás, set ég engifer í teið.

Notice that after the fronted clause, it becomes set ég, not ég set.

That is a classic Icelandic word-order rule.

Yes, but the meaning changes slightly.

• í teið = into the tea
• í te = into tea / into some tea

The version with teið sounds more specific, as if you mean a particular cup or serving of tea. The version without the article is more indefinite or general.