Когда у дочки озноб, мама даёт ей тёплый чай и смотрит на градусник каждые полчаса.

Here it means when/whenever in a general, repeated situation.

So this sentence is not talking about one specific moment only. It means something like:

Whenever the daughter has chills, the mother gives her warm tea and checks the thermometer every half hour.

Russian often uses когда + present tense for this kind of habitual meaning.

Russian very often expresses physical conditions with the pattern:

у + genitive + noun

So у дочки озноб literally means something like at the daughter there is chills, but naturally it means the daughter has chills.

This pattern is very common:

• у меня температура = I have a temperature / fever
• у него кашель = he has a cough
• у ребёнка насморк = the child has a runny nose

Using иметь in these everyday health situations would sound unnatural.

Because after у in this construction, Russian uses the genitive case.

The dictionary form is дочка.
In the genitive singular, it becomes дочки.

So:

• дочка = daughter
• у дочки = the daughter has / at the daughter

Yes. Дочь is the basic word daughter.
Дочка is a diminutive / affectionate form.

In many family contexts, дочка sounds warm, natural, and very common in everyday speech. It does not always mean little daughter literally, but it often has a more affectionate tone than дочь.

Ей is the dative singular form of она.

The verb давать / дать uses:

• the thing given in the accusative
• the person receiving it in the dative

So:

• даёт чай = gives tea
• даёт ей чай = gives her tea

You could also say: мама даёт дочке тёплый чай

There, дочке is also dative.

It does become the accusative, but for a masculine inanimate noun, the accusative looks the same as the nominative.

So:

• nominative: тёплый чай
• accusative: тёплый чай

That is why there is no visible change here.

Compare with an animate masculine noun, where you would see a difference:

• nominative: новый врач
• accusative: нового врача

Because the verb смотреть normally takes на + accusative for the thing someone is looking at.

So the pattern is:

смотреть на кого? на что?

Examples:

• смотреть на небо = look at the sky
• смотреть на ребёнка = look at the child
• смотреть на градусник = look at the thermometer

In English, look also usually needs at, and Russian needs на here.

It is accusative singular because it comes after на with смотреть на.

The noun is masculine and inanimate, so its accusative form is the same as the nominative:

• nominative: градусник
• accusative: градусник

So the form does not visibly change, but the case is still accusative.

Каждые полчаса is the normal idiomatic way to say every half hour.

Time intervals after каждый are often expressed this way:

• каждый час = every hour
• каждые два часа = every two hours
• каждые полчаса = every half hour

With полчаса, Russian normally uses каждые in this expression. So каждое полчаса is not standard.

It is best to learn каждые полчаса as a fixed, natural phrase.

Because Russian present tense is often used for habitual or general actions, just like the English simple present.

So this does not mean the action is happening only right now. It means this is what the mother usually does in that situation.

Compare:

• Когда ребёнок болеет, мама волнуется.
• Если у него температура, она вызывает врача.

These are general truths or repeated actions, so present tense is normal.

Because Когда у дочки озноб is a subordinate clause, and Russian separates it from the main clause with a comma.

So the structure is:

Когда у дочки озноб, мама даёт ей тёплый чай...

This is the same basic idea as in English: When the daughter has chills, the mother gives her warm tea...

Russian punctuation requires that comma.