Learn Hardcore Croatian: Učiteljica nam daje dodatne vježbe kad se spremamo za ispit. - The teacher gives us additional exercises when we are preparing for the exam.

nam is a clitic (an unstressed short word) and in Croatian, clitics typically stand in second position in the clause.
The first stressed element is Učiteljica, so nam comes right after it:
Učiteljica nam daje…

You could technically say Učiteljica daje nama dodatne vježbe, but that sounds more marked/emphatic (stressing to us), and the original word order with the clitic nam is the most natural.

Which tense and person is daje, and from which verb does it come?

Daje is:

3rd person singular
present tense
of the verb davati (to give, imperfective, ongoing/habitual).

Conjugation (present of davati):

ja dajem – I give
ti daješ – you give (sg)
on/ona/ono daje – he/she/it gives
mi dajemo – we give
vi dajete – you give (pl/formal)
oni/one/ona daju – they give

The imperfective verb davati is used here because it describes a repeated or habitual action:

Učiteljica nam daje dodatne vježbe…
= The teacher *gives us extra exercises (whenever this situation happens)*.

The related perfective verb is dati (to give once, as a completed act), with a different present: dam, daš, da, damo, date, daju — but that present is less common and usually refers to future-like events, not a general habit.

Why is it dodatne vježbe and not something like dodatni vježbe?

This is about agreement in gender, number, and case between the adjective and the noun.

vježba = exercise (singular, feminine)
vježbe = exercises (plural, feminine, nominative or accusative)
dodatni = additional (masculine form)
dodatne = additional (feminine plural accusative / nominative)

In daje dodatne vježbe, dodatne vježbe is the direct object of the verb, so it is in the accusative plural feminine:

noun: vježb-e (fem., acc. plural)
adjective: dodatn-e (fem., acc. plural)

So dodatni vježbe would be wrong because dodatni is masculine, but vježbe is feminine plural; they must match: dodatne vježbe.

What does kad se spremamo literally mean, and what is the role of se?

Kad se spremamo literally means when we are preparing (ourselves).

kad = when
spremamo = we prepare / we are preparing (1st person plural present of spremati)
se = reflexive pronoun (ourselves, yourself, etc., depending on person)

Spremati se is a reflexive verb, meaning to get ready; to prepare oneself. In many contexts, the se is necessary:

spremati (nešto) = to prepare something
- Spremamo ručak. = We are preparing lunch.
spremati se (za nešto) = to prepare oneself for something / to get ready for something
- Spremamo se za ispit. = We are preparing (ourselves) for the exam.

So se here shows that the subject is preparing themselves, not some other object.

Can I say kada instead of kad? Is there any difference?

Yes, you can say kada instead of kad:

Kad se spremamo za ispit…
Kada se spremamo za ispit…

Both mean when we are preparing for the exam.

Differences:

kada is a bit more formal or careful.
kad is shorter and very common in everyday speech.

In most contexts they are interchangeable; using kad here is completely natural and normal.

Why is it za ispit and not na ispit? What does za mean here?

Za + accusative often means for (with the sense of “in preparation for / in view of”).

za ispit = for the exam (in preparation for the exam)

In this sentence:

spremamo se za ispit
= we are preparing for the exam

Na ispit (also with accusative) has a different meaning:

ići na ispit = to go to the exam (to attend/take it)

So:

spremati se za ispit – to prepare for the exam
ići na ispit – to go to the exam / to sit the exam

That’s why za ispit is used here.

What case is ispit in, and why doesn’t it change form?

Ispit is in the accusative singular after za.

Base form (nominative): ispit (exam), masculine, inanimate
Accusative singular (for inanimate masculine nouns): ispit (same form as nominative)

In Croatian, inanimate masculine nouns usually have the same form in nominative and accusative singular:

vidim stol (I see a table) – same as nominative stol
spremamo se za ispit – same as nominative ispit

So it did change grammatically to accusative, but the form just happens to look identical to the nominative.

Why is the present tense used in both parts: daje and spremamo se? Could I say something like kad ćemo se spremati?

The present tense in Croatian is commonly used for:

General truths / habits / repeated actions, and
Time clauses with “kad(kada)” referring to a general, repeated, or future situation.

In this sentence:

Učiteljica nam daje dodatne vježbe kad se spremamo za ispit.
= The teacher gives us extra exercises when we are preparing for the exam.
(meaning: whenever that situation occurs, as a habit)

Both daje and spremamo se are present tense, showing a habitual relationship.

You might see kad ćemo se spremati in other contexts, but that usually has a more specific future sense, like:

Kad ćemo se spremati za ispit? = When will we be preparing for the exam? (asking about schedule)

For a general rule or typical situation, the normal pattern is present + kad + present, as in the original sentence.

Can I omit se and say kad spremamo za ispit?

No, kad spremamo za ispit is not correct in this meaning.

You have two different verbs:

spremati (nešto) – to prepare something
- Spremamo ručak. = We are preparing lunch.
- Spremamo ispit. (possible, but more like “We are preparing the exam” from a teacher’s perspective, not students studying)
spremati se (za nešto) – to prepare (oneself) for something / to get ready for something
- Spremamo se za ispit. = We are preparing (ourselves) for the exam / we are getting ready for the exam.

In the context of students studying for an exam, you need the reflexive form spremati se:

kad se spremamo za ispit = when we are preparing for the exam

Without se, the verb would need a direct object (spremamo nešto), and za ispit alone isn’t that direct object, so the sentence would be ungrammatical or would change meaning.