Learn Hardcore Portuguese (Portugal): Durante o desfile de moda, cada marca mostrou chapéus elegantes. - During the fashion show, each brand showed elegant hats.

durante is a preposition meaning “throughout” or “during” when followed by a noun or noun phrase. It marks a time span.
By contrast, enquanto is a conjunction meaning “while” and must introduce a clause with its own verb.
Examples:

Durante o verão, choveu pouco.
Enquanto chovia, ficámos em casa.

You cannot say durante chovia because durante needs a noun, not a verb.

Why do we say desfile de moda instead of desfile moda? What role does de play here?

The preposition de links two nouns in a “genitive” or “of” relationship.

desfile = show
moda = fashion
So desfile de moda literally means “show of fashion,” i.e. “fashion show.”
If you drop de, you break that link: desfile moda is ungrammatical.

Why is there an article o before desfile, but not before cada marca?

You use the definite article o with desfile because you’re referring to a specific event (the fashion show).
By contrast, cada (“each”) already gives a sense of specificity to marca, so Portuguese does not combine cada with another article.
Wrong: cada a marca or cada as marcas
Correct: cada marca (each brand)

What does cada mean, and does it change for gender or number?

cada means “each” or “every.”
It is always singular and agrees in gender with the noun it modifies (but never changes for number).
• Feminine: cada marca, cada casa
• Masculine: cada homem, cada dia

You cannot say cada marcas – it must remain singular.

Why is the verb mostrou singular even though chapéus is plural?

The subject of the verb is cada marca, which is grammatically singular.
Portuguese verb agreement follows the subject, not the object.

Subject: cada marca (singular)
Verb: mostrou (3rd person singular)
Object: chapéus (plural)

If the subject were plural (e.g. as marcas), you would say mostraram.

Why is the adjective elegantes placed after chapéus, and how does adjective agreement work?

In Portuguese, descriptive adjectives most commonly follow the noun:

chapéus elegantes (elegant hats)
Agreement rules:
Number: chapéu → chapéus, elegante → elegantes
Gender: hats (masculine plural) → elegantes (masculine plural form is identical to feminine plural for this adjective)

The order (noun + adjective) is standard for neutral/descriptive emphasis.

Why is there no preposition between mostrar and chapéus elegantes? Is mostrar transitive?

Yes, mostrar is a transitive verb that can take a direct object without a preposition:

Eu mostro o livro.
Ela mostra os chapéus.

If you specify to whom you show something, you add a (indirect object):

Ela mostra os chapéus ao público.

Could we say elegantes chapéus instead of chapéus elegantes? Would it change the emphasis or meaning?

You can say elegantes chapéus, but it’s less neutral and often more poetic or emphatic.

chapéus elegantes = straightforward description
elegantes chapéus = highlights the elegance, stylistic choice
In everyday speech and writing, noun-first order is preferred for simple descriptions.