Mér er illt í tönninni í dag.

Because this is an impersonal Icelandic pattern for physical sensations.

Icelandic often says things like:

• Mér er kalt = I am cold
• Honum er illt = He is in pain

So the person experiencing the feeling goes in the dative case, not the nominative. That is why you get mér instead of ég.

A very literal way to think about Mér er illt is To me it is painful.

In this kind of impersonal construction, the adjective is normally put in the neuter singular by default.

So:

• mér er illt
• henni er kalt
• þeim er heitt

The adjective is not agreeing with mér or with tönninni here. It stays in that default neuter singular form because the sentence does not have an ordinary nominative subject that the adjective would agree with.

Here illt does not just mean bad in a general sense. In this construction, vera illt means something like:

• to hurt
• to be painful
• to be sore

So Mér er illt í tönninni means that there is pain in the tooth area — in natural English, my tooth hurts or I have a toothache.

Icelandic often expresses pain by saying it is painful in a body part:

• í höfðinu = in the head
• í bakinu = in the back
• í maganum = in the stomach
• í tönninni = in the tooth

So the structure is very natural in Icelandic: someone-DAT + er illt + í + body part.

It is basically locating the pain.

Because the preposition í takes the dative when it describes location or state, and that is what is happening here.

There is no movement into the tooth. The pain is simply located there.

So after í you need the dative:

• í tönninni = in the tooth

By comparison:

• tönnin is nominative
• tönnina is accusative
• tönninni is dative

With body parts, Icelandic often does not use a possessive like English does when the owner is obvious from context.

So instead of saying my tooth, Icelandic very naturally says the tooth:

• Mér er illt í tönninni = literally It hurts me in the tooth

Since mér already tells you who is affected, it is understood that this is my tooth.

You can say í tönninni minni, but that is usually more emphatic and often unnecessary.

Yes, tönninni is singular, so it refers to one tooth.

If you wanted to talk about more than one tooth, you would use the plural, for example:

• Mér er illt í tönnunum = My teeth hurt

That said, in everyday speech, a singular body-part expression can sometimes sound natural even when the speaker is just talking about tooth pain in general.

Yes. Icelandic word order is fairly flexible, especially with time expressions.

The sentence you have is a very normal neutral order:

• Mér er illt í tönninni í dag.

But you could also say:

• Í dag er mér illt í tönninni.

That version emphasizes today a bit more.

The main thing to remember is that in a main clause, the finite verb usually stays in second position.

Not in the way English uses it.

English often needs a subject even when it does not really refer to anything:

• It hurts
• It is cold

Icelandic often allows these impersonal expressions without an explicit dummy subject. So Mér er illt is a complete sentence by itself.

That is why you do not normally add það here.

Yes. That is very natural too.

Both illt and vont can be used for pain, but many speakers use vont very often in everyday speech:

• Mér er vont í tönninni = My tooth hurts

Illt is also correct and natural, but vont may sound a bit more colloquial or everyday to many learners' ears.

Yes. A very common alternative is to use a noun phrase such as:

• Ég er með tannpínu = I have a toothache

That is often a straightforward everyday way to express the same idea.

So these are all natural, though not exactly identical in focus:

• Mér er illt í tönninni = My tooth hurts / I have pain in the tooth
• Mér er vont í tönninni = My tooth hurts
• Ég er með tannpínu = I have a toothache