Mio fratello dice che ho un dente sensibile al freddo.

In Italian, the definite article is usually left out with singular family members when they are not modified.

So you normally say:

• mio fratello = my brother
• mia sorella = my sister
• mio padre = my father

But you usually use the article with:

• plural family members: i miei fratelli
• modified family members: il mio fratello maggiore
• some special forms such as la mia mamma

So mio fratello is the normal form here.

Because ho still refers to I.

The sentence means:

• My brother says that I have a tooth sensitive to cold.

So the main clause subject is:

• Mio fratello = my brother

But in the clause after che, the subject changes to:

• io = I

That is why Italian uses:

• ho = I have

Compare:

• Mio fratello dice che ho un dente sensibile al freddo.
  = My brother says that I have a tooth sensitive to cold.

• Mio fratello dice che ha un dente sensibile al freddo.
  = My brother says that he has a tooth sensitive to cold.

Che here means that and introduces a subordinate clause.

• Mio fratello dice = My brother says
• che ho un dente sensibile al freddo = that I have a tooth sensitive to cold

In English, that is often optional:

• My brother says I have...
• My brother says that I have...

In Italian, che is normally required in this kind of sentence.

Here un is the indefinite article, so it means a:

• un dente = a tooth

It does not strongly mean one tooth here. It just introduces a singular noun that is not specific yet.

If you wanted to emphasize the number, context would do that, or you might stress un in speech.

Also note that dente is a masculine singular noun, so it takes un.

Because sensibile is an adjective that agrees with dente.

• dente is masculine singular
• so the adjective must also be masculine singular
• that form is sensibile

Examples of agreement:

• un dente sensibile = a sensitive tooth
• una gengiva sensibile = a sensitive gum
• denti sensibili = sensitive teeth

This adjective belongs to a group where the singular form is often the same for masculine and feminine:

• sensibile (masculine singular)
• sensibile (feminine singular)
• sensibili (plural)

Al is the contraction of a + il:

• a + il = al

So literally:

• sensibile al freddo = sensitive to the cold

This is a very normal Italian structure. With sensibile, Italian often uses a to express what something reacts to:

• sensibile al caldo = sensitive to heat
• sensibile alla luce = sensitive to light
• sensibile al rumore = sensitive to noise

So al freddo is the standard way to say to cold.

Here freddo is being used as a noun, meaning cold.

So:

• il freddo = the cold

In the phrase al freddo, it means to the cold.

Italian often uses adjectives as nouns in this way:

• il caldo = heat / the heat
• il freddo = cold / the cold
• il bello = beauty / what is beautiful

So in this sentence, freddo is not describing dente directly. It is the thing the tooth is sensitive to.

Yes. This phrase is general and can be used for many things that react badly or strongly to cold.

For example:

• Ho la pelle sensibile al freddo. = I have skin that is sensitive to cold.
• Le mie mani sono sensibili al freddo. = My hands are sensitive to cold.

In your sentence, it is especially natural because people often talk about a tooth reacting painfully to cold food or drinks.

That is just a difference between the two languages.

English says:

• sensitive to cold

Italian prefers:

• sensibile al freddo

Italian often uses an article where English does not. In this case, al already includes il, so the article is built into the phrase.

This is very common and natural, not something unusual about this sentence.

It is normal word order.

The sentence is built like this:

• Mio fratello = subject
• dice = verb
• che = connector
• ho un dente sensibile al freddo = subordinate clause

Inside the subordinate clause:

• ho = verb
• un dente = object
• sensibile al freddo = adjective phrase describing dente

So the structure is very straightforward and natural for everyday Italian.

The sentence you have is indirect speech:

• Mio fratello dice che ho un dente sensibile al freddo.

If you changed it to direct speech, it might be:

• Mio fratello dice: Ho un dente sensibile al freddo.
  = My brother says: I have a tooth sensitive to cold.

Or, if he is talking about you, direct speech could be:

• Mio fratello dice: Hai un dente sensibile al freddo.
  = My brother says: You have a tooth sensitive to cold.

So indirect speech keeps the reporting structure with che, and the verb form depends on who the subject is in the reported clause.