Hvað sem kennarinn spyr, reyni ég að svara á íslensku.

Hvað sem is a set pattern meaning whatever or no matter what.

So:

• hvað = what
• sem does not really have a neat one-word English equivalent here; it helps build the meaning whatever / no matter what

If you said just hvað, it would simply mean what, not whatever.

Yes. It is a subordinate clause meaning whatever the teacher asks.

A useful way to understand it is:

• kennarinn spyr = the teacher asks
• hvað sem supplies the thing being asked

So hvað functions as the object of spyr.

Because kennarinn means the teacher, while kennari means a teacher or just teacher.

Icelandic usually adds the definite article to the end of the noun:

• kennari = teacher
• kennarinn = the teacher

That -inn ending is the attached definite article.

Spyrja is the infinitive, meaning to ask.
Spyr is the present tense form, asks.

So:

• að spyrja = to ask
• ég spyr = I ask
• kennarinn spyr = the teacher asks

This is very normal in Icelandic: the dictionary form ends in -a, but the finite present-tense form often does not.

This is because Icelandic is a verb-second language in main clauses.

The first slot in the sentence is occupied by the whole subordinate clause:

Hvað sem kennarinn spyr

After that, the finite verb of the main clause comes next:

reyni

Then the subject:

ég

So:

• Ég reyni að svara á íslensku. = neutral word order
• Hvað sem kennarinn spyr, reyni ég að svara á íslensku. = the subordinate clause comes first, so the verb moves before the subject in the main clause

This is one of the most important word-order patterns in Icelandic.

Because reyna normally takes að + infinitive.

So:

• reyna að gera eitthvað = to try to do something
• reyni að svara = try to answer

This is similar to English try to answer, except Icelandic uses að before the infinitive.

Because that is the normal Icelandic way to say in a language.

Examples:

• á íslensku = in Icelandic
• á ensku = in English
• á dönsku = in Danish

So even though English uses in, Icelandic usually uses á in this expression. It is best learned as a fixed pattern.

It is the form used in the fixed expression á íslensku.

For learners, the most helpful approach is usually to memorize the whole pattern:

• á íslensku
• á ensku
• á þýsku

In other words, do not worry too much at first about forcing a word-for-word English-style analysis. The whole phrase simply means in Icelandic.

Not necessarily. Svara can take an object, often in the dative, for example:

• svara spurningunni = answer the question

But here the sentence does not need to repeat what is being answered, because that idea is already understood from hvað sem kennarinn spyr.

So the sentence focuses on how the speaker answers:

• whatever the teacher asks,
• I try to answer in Icelandic

Because the subordinate clause comes first, and the main clause follows after it.

So the comma separates:

• Hvað sem kennarinn spyr = subordinate clause
• reyni ég að svara á íslensku = main clause

This is very similar to English punctuation in a sentence like:

Whatever the teacher asks, I try to answer in Icelandic.

Not in this sentence.

• hvað sem
  • clause = whatever / no matter what
• hvað sem er usually means anything

So:

• Hvað sem kennarinn spyr = Whatever the teacher asks
• hvað sem er = anything

They are related, but they are not interchangeable here.

A very literal breakdown is:

• Hvað sem = whatever
• kennarinn = the teacher
• spyr = asks
• reyni ég = try I
• að svara = to answer
• á íslensku = in Icelandic

A more natural English rendering is:

Whatever the teacher asks, I try to answer in Icelandic.

So the biggest things to notice are:

• hvað sem = whatever
• verb-second word order: reyni ég
• reyna að + infinitive
• á íslensku = in Icelandic