Learn Hardcore French: Elle écrit dans son journal que sa fièvre a presque disparu, que sa toux est plus rare et que cette expérience l’a rendue plus prudente. - She writes in her diary that her fever has almost disappeared, that her cough is rarer, and that this experience has made her more cautious.

In French, using the present to report what someone writes (or says, thinks, etc.) is very common, even if the content itself refers to past events.

Elle écrit dans son journal que… = She writes in her diary that…
This can mean “she is (currently) writing” or “she (generally) writes / has written”.
Inside that, you can use past tenses for events:
sa fièvre a presque disparu = her fever has almost disappeared (completed change).

So the main verb (écrit) is a present of narration / reporting, while the subordinate clauses can have whatever tense fits their own time frame.

Why is it dans son journal and not something like sur son journal or à son journal?

With journal (intime) meaning “diary”, the normal preposition in French is dans:

écrire dans son journal = write in one’s diary (inside its pages).

Sur would suggest writing on the surface of something (e.g. sur le mur = on the wall), and à son journal would sound like writing to the diary as if it were a person, which is not the standard way to phrase it.

Why is it sa fièvre a presque disparu in the passé composé and not sa fièvre disparaissait in the imperfect?

The passé composé (a disparu) is used for completed events or changes:

sa fièvre a presque disparu = her fever has almost disappeared (the change from “fever” to “no fever” is basically done).

The imparfait (disparaissait) would describe an ongoing or repeated process in the past, something like:

sa fièvre disparaissait = her fever was disappearing / used to disappear (process, not result).

Here, the idea is that the fever has (almost) reached the end of its disappearance process, so passé composé fits better.

Why is the adverb placed as a presque disparu and not presque a disparu?

Most short adverbs like presque, déjà, encore, souvent, toujours go between the auxiliary and the past participle:

a presque disparu
a déjà mangé
a souvent voyagé

Putting presque before the auxiliary (presque a disparu) is ungrammatical in standard French. The default order is:

subject + auxiliary + adverb + past participle

What does sa toux est plus rare literally mean, and is rare common with toux?

Literally:

sa toux est plus rare = her cough is rarer / she coughs less often.

In this kind of health context, rare is commonly used about symptoms to mean “infrequent” rather than “uncommon in the world”. So:

sa toux est rare = she coughs rarely.
sa toux est plus rare = she coughs less often than before.

You could also say sa toux est moins fréquente, but plus rare is natural and idiomatic.

Why is it cette expérience l’a rendue plus prudente and not l’a rendu plus prudente?

The direct object pronoun l’ stands for elle (the woman). With avoir as the auxiliary, the past participle rendue agrees in gender and number with a direct object placed before the verb:

COD (direct object) l’ = elle, feminine singular
Auxiliary: a
Past participle: rendu → rendue (feminine singular agreement)

So:

Cette expérience a rendu elle plus prudente → after pronoun replacement:
Cette expérience l’a rendue plus prudente.

If the direct object came after the verb (Cette expérience a rendu Marie plus prudente), the participle would stay rendu, because there is no preceding direct object to trigger agreement.

Who does l’ refer to in l’a rendue, and can it refer to cette expérience?

Here, l’ refers to elle (the woman), not to cette expérience.

Subject: cette expérience (this experience)
Direct object: elle → l’ (before the verb)
Verb: a rendue (has made)

So the structure is:

Cette expérience (subject) l’ (her) a rendue plus prudente.

It cannot refer to cette expérience, because cette expérience is already the subject and it is feminine singular; there would be no reason to replace it with a pronoun here.

Why is it rendue plus prudente (both feminine) and not rendu plus prudent?

Two agreements happen:

rendue agrees with the direct object (the woman, elle, feminine singular) because the object pronoun l’ comes before the auxiliary a.
prudente agrees with the person being described (again, elle, feminine singular).

So:

Masculine: Il l’a rendu plus prudent.
Feminine: Elle l’a rendue plus prudente.

If the person were male, you would see rendu plus prudent instead.

Why is the possessive adjective son used in son journal, even though the owner is a woman (elle)?

In French, son / sa / ses agree with the gender and number of the noun possessed, not with the gender of the possessor.

journal is masculine singular → son journal
fièvre is feminine singular → sa fièvre
toux is feminine singular → sa toux

So a woman says:

son journal, sa fièvre, sa toux
and a man would also say exactly the same, because it depends on journal / fièvre / toux, not on il / elle.

Why is que repeated three times: que sa fièvre…, que sa toux…, que cette expérience…? Could you omit some of them?

In French, each subordinate clause introduced by que normally keeps its own que, even when the clauses are coordinated:

Elle écrit… que A, que B et que C.

You cannot omit the second and third que the way you sometimes can in English. So:

Correct: …qu’elle va mieux, que sa toux est plus rare et que cette expérience l’a rendue plus prudente.
Incorrect: …qu’elle va mieux, sa toux est plus rare et cette expérience l’a rendue plus prudente.

Could we say qu’elle est plus prudente instead of que cette expérience l’a rendue plus prudente?

Grammatically, qu’elle est plus prudente is fine, but the meaning is slightly different:

qu’elle est plus prudente: simply states she is more cautious now (no explicit cause).
que cette expérience l’a rendue plus prudente: explicitly says this experience has made her more cautious (cause and result).

The original sentence emphasizes that the experience caused a change in her behavior.