Learn Hardcore Croatian: Hodajući kroz šumu, ona sluša ptice i duboko diše. - Walking through the forest, she listens to the birds and breathes deeply.

Hodajući is a present adverbial participle, in Croatian called glagolski prilog sadašnji.
It is formed from the 3rd person plural present tense of a verb:

3rd person plural: oni hodaju
remove -u and add -ći → hodajući

It roughly corresponds to English walking in sentences like Walking through the forest, she…

Why is hodajući used instead of a normal verb like hoda or a clause like dok hoda?

Hodajući makes the action of walking into a background action that happens at the same time as the main actions (sluša, diše).
You could also say:

Dok hoda kroz šumu, ona sluša ptice i duboko diše.

This is correct, but a bit more explicit and clause-like.
The version with hodajući is more compact and stylistically a bit more elegant, similar to English Walking through the forest, she listens…

Does hodajući agree with ona in gender or number?

No. The present adverbial participle hodajući is invariable: it does not change for gender, number, or case.
Whether the subject is ona, on, oni, or one, the form stays hodajući:

Hodajući kroz šumu, on sluša ptice.
Hodajući kroz šumu, oni slušaju ptice.

Why is there a comma after Hodajući kroz šumu?

In Croatian, an introductory participial or adverbial phrase is normally separated from the main clause by a comma, just like in English:

Hodajući kroz šumu, ona sluša ptice i duboko diše.

The comma marks the boundary between the background action (hodajući kroz šumu) and the main clause (ona sluša… i duboko diše).

Why is the preposition kroz followed by šumu, not šuma?

The preposition kroz (through) always takes the accusative case.
Šuma is a feminine noun:

nominative singular: šuma (subject form)
accusative singular: šumu (after kroz)

So you must say kroz šumu = through the forest.

Could you say Hodajući šumom instead of Hodajući kroz šumu? Is there any difference?

Yes, Hodajući šumom is also correct. Here šumom is in the instrumental case, without a preposition.
Nuance:

kroz šumu – slightly emphasizes movement through the forest from one side to another.
šumom – focuses more on being/moving in the forest in general.

Both can work in this sentence; the difference is subtle and often stylistic.

Why is ona (she) expressed? Could you leave it out?

In Croatian, subject pronouns are usually dropped because the verb ending already shows the person:

(Ona) sluša ptice. → Sluša ptice.

Here ona is not strictly necessary; Hodajući kroz šumu, sluša ptice i duboko diše. is grammatically correct.
Using ona can add a bit of emphasis or clarity if needed, but in context it is often omitted.

Why is sluša used and not čuje? Don’t both relate to hearing?

Yes, but they differ like English listen vs hear:

slušati = to listen (an intentional action, paying attention)
čuti = to hear (more passive perception)

So ona sluša ptice means she is actively listening to the birds, not just accidentally hearing them.

What case and number is ptice, and why that form?

Ptice here is accusative plural of ptica (bird):

nominative singular: ptica
accusative singular: pticu
nominative plural: ptice
accusative plural: ptice

For many feminine nouns, nominative and accusative plural have the same form -e.
Because slušati takes a direct object (whom? what?), it needs the accusative: sluša ptice = she listens to (the) birds.

Can word order change? For example, could you say Ona sluša ptice i diše duboko or Duboko diše?

Yes, Croatian word order is flexible. All of these are correct:

Ona sluša ptice i duboko diše.
Ona sluša ptice i diše duboko.
Sluša ptice i duboko diše.

Duboko diše vs diše duboko: both are grammatical; duboko diše (adverb before the verb) is slightly more neutral and common in this type of description, but the difference is small.

Why are the verbs sluša and diše in the simple present, even though English uses a progressive form (is listening, is breathing)?

Croatian does not have a special progressive tense. The simple present is used both for:

general/habitual actions
actions happening right now

So ona sluša ptice i duboko diše can correspond to English she listens / is listening and she breathes / is breathing, depending on context. Here, the context (with hodajući) makes it clear these are actions happening at the same time.