Mér er illt í tönninni þegar ég drekk kaffi.

Because this is an impersonal Icelandic expression.

In English, we usually say something like my tooth hurts. In Icelandic, a very common pattern is:

mér er illt = I am in pain / it hurts

Here, the person experiencing the pain is put in the dative case, so ég becomes mér.

So:

• ég = I (nominative)
• mér = to me / for me (dative)

This same pattern appears in other common expressions:

• Mér er kalt. = I am cold.
• Honum er heitt. = He is hot.
• Henni er illt í bakinu. = Her back hurts.

Here illt means something like painful, hurts, or sore.

It is the neuter singular form of the adjective illur. In this kind of impersonal construction, Icelandic often uses the neuter singular as a default form.

So:

• mér er illt = I am in pain / it hurts
• not literally I am bad

This is important, because illur / illt can mean bad in some contexts, but in this expression it refers to pain.

Also, this is not the adverb illa.
For example:

• Mér líður illa. = I feel bad / sick.
• Mér er illt í tönninni. = My tooth hurts.

Because Icelandic often describes localized pain with the pattern:

mér er illt í + body part

So the idea is:

• mér er illt í bakinu = my back hurts
• mér er illt í höfðinu = my head hurts
• mér er illt í tönninni = my tooth hurts

English usually makes the body part the grammatical subject: my tooth hurts.

Icelandic instead often treats it more like there is pain in the tooth for me.

So í is very natural and idiomatic here.

Because Icelandic very often uses the definite article with body parts when it is already obvious whose body part is meant.

Since mér tells us who is experiencing the pain, Icelandic does not need to say my explicitly. So:

• Mér er illt í tönninni. = My tooth hurts.

This is extremely natural.

You can say í tönninni minni, but that is usually more explicit than necessary, and may sound emphatic unless there is a reason to stress my tooth rather than someone else’s.

Tönninni is dative singular definite.

That happens because the preposition í can take:

• accusative for motion into something
• dative for location in something

Here there is no movement. The pain is located in the tooth, so Icelandic uses the dative:

• í tönninni

So the grammar is:

• í
  • dative
• because this is a location/state meaning, not motion

Because tönn is an irregular noun.

Its singular forms are:

• nom. tönn
• acc. tönn
• dat. tönn
• gen. tannar

With the definite article, the dative singular becomes:

• tönninni

So even though tönninni may look complicated at first, it is just:

tönn + definite article + dative ending

This is one of those nouns that learners usually just have to get used to by exposure.

Because drekk is the 1st person singular present tense of drekka (to drink).

The subject is ég, so the verb must match that:

• ég drekk = I drink
• þú drekkur = you drink
• hann/hún/það drekkur = he/she/it drinks
• við drekkum = we drink

So in þegar ég drekk kaffi, the form drekk is exactly the right one for ég.

Because kaffi here is being used as a general substance noun: coffee in general, not the coffee.

So:

• ég drekk kaffi = I drink coffee
• ég drekk kaffið = I drink the coffee

In the sentence þegar ég drekk kaffi, the meaning is general: drinking coffee as a type of drink, not a specific cup already identified in the conversation.

Usually, with the present tense like this, it often has a general/habitual meaning:

whenever I drink coffee, my tooth hurts

So even though English may translate it as when I drink coffee, the sense is often broader than one single occasion.

In other words, the sentence usually suggests a recurring pattern, not just one specific moment.

Yes. You can also say:

Þegar ég drekk kaffi, er mér illt í tönninni.

That is also correct and natural.

A useful thing to notice is the word order in the main clause after a fronted subordinate clause:

• Þegar ég drekk kaffi, er mér illt í tönninni.

The verb er comes before mér. That is normal Icelandic verb-second behavior in the main clause.

So both of these are fine:

• Mér er illt í tönninni þegar ég drekk kaffi.
• Þegar ég drekk kaffi, er mér illt í tönninni.

This is a very natural and idiomatic way to say it.

It is especially good if you want to describe pain in a specific body part.

A few alternatives are possible, for example:

• Ég finn til í tönninni þegar ég drekk kaffi.
  = I feel pain in my tooth when I drink coffee.

• Ég fæ tannverk þegar ég drekk kaffi.
  = I get toothache when I drink coffee.

But your original sentence is excellent Icelandic, and it teaches a very useful pattern:

mér er illt í + body part