Learn Hardcore Portuguese (Portugal): A veterinária disse que a gata deve usar a trela na sala de espera. - The vet said that the cat should use the leash in the waiting room.

a veterinária = the vet is a woman. Male would be o veterinário.
a gata = the cat is specifically female. Male (or default when sex is unknown) is o gato.
Examples:
A veterinária disse que o gato… (female vet, male/unspecified cat)
O veterinário disse que a gata… (male vet, female cat)

Does deve mean “must” or “should” here?

In European Portuguese, deve (from dever) often suggests a recommendation or obligation that’s softer than an absolute rule—so “should” or “ought to.” A stronger “must” is usually expressed with tem de (or tem que, also heard):

Softer: A gata deve usar a trela… (The cat should use a leash…)
Stronger: A gata tem de usar a trela… (The cat must use a leash…)

Note: dever can also mean probability (“must” as in “probably”): Ela deve estar atrasada (She must be/probably is late). In your sentence, context makes it obligation/advice.

Why is it indicative after disse que and not the subjunctive?

Dizer que introduces reported information, so the indicative is normal: disse que a gata deve… If you want to report a command/request, Portuguese often uses:

dizer para
- infinitive: A veterinária disse para pôr a gata à trela.
Or a verb that triggers the subjunctive (e.g., mandar que): A veterinária mandou que a gata estivesse à trela (formal/less common in everyday speech).

Should it be disse à gata instead of disse que a gata? What’s the difference?

disse que = “said that” (introduces the content of what was said).
disse à gata = “said to the cat” (indicates the person/animal addressed).
The grave accent in à shows the contraction of a + a (“to the”).

Your sentence focuses on the message, so disse que is right. If you want the addressee too: A veterinária disse à dona da gata que ela deve usar a trela… (said to the cat’s owner that she/it should use a leash).

Is usar a trela idiomatic, or is there a more natural way to say it in Portugal?

It’s understandable, but the most idiomatic EP phrasing is:

estar à trela or andar à trela (be/walk on a leash).

Natural clinic wording: Os animais devem estar à trela na sala de espera.
For “put the leash on”: pôr a trela no cão/na gata or pôr o cão/a gata à trela.

What’s the difference between trela, coleira, and arnês/peitoral?

trela = the leash (the strap/lead you hold).
coleira = the collar (around the neck).
arnês/peitoral = a harness.

Brazil note: in Brazil, guia is common for “leash”; coleira is mainly “collar.” In Portugal, “leash” is trela.

What does na mean in na sala de espera?

na = contraction of em + a = “in the/at the” for a feminine noun:

na sala (in the room), na caixa (in the box).
Masculine uses no (em + o): no carro (in the car).

Why is it sala de espera and not sala da espera?

Portuguese forms many compound nouns with de to show purpose/type, without an article:

sala de espera (waiting room), quarto de banho (bathroom), sala de jantar (dining room).

Using da would imply a specific, possessed “waiting” (odd here). So sala de espera is the standard compound.

Could I drop the articles and say just gata or trela?

Generally no. Portuguese uses definite articles with specific, countable nouns much more than English:

Natural: a gata, a trela.
Without article sounds generic or incomplete.

If you mean “any leash,” use the indefinite: uma trela (e.g., A gata deve usar uma trela nova?).

Why is the subordinate verb in the present (deve) when the main verb is past (disse)?

Reported speech in Portuguese doesn’t require tense backshift. Use the present when the statement is still valid now:

A veterinária disse que a gata deve usar a trela (rule still applies). If it was only an obligation at that past time, you can backshift:
A veterinária disse que a gata devia usar a trela (it was required then).

Can I move na sala de espera to another position?

Yes, with normal word-order flexibility and commas for clarity:

Na sala de espera, a gata deve usar a trela.
A veterinária disse que, na sala de espera, a gata deve usar a trela.

Be aware: A veterinária disse na sala de espera que… means “The vet said (this) in the waiting room,” i.e., it modifies where she spoke, not where the leash must be used.

How would I replace a trela with a pronoun?

Most natural: attach it to the infinitive: A gata deve usá-la na sala de espera.
Note: final -r of usar drops and you add -la, with an accent: usá-la.
With ter de both are heard in EP: A gata tem de a usar / A gata tem de usá-la.

If I want to use “put,” is it por or pôr?

The verb is pôr (with circumflex). por (no accent) is the preposition “by/through/for.”
Examples:

pôr a trela na gata / pôr a gata à trela (put the leash on the cat).
Not: ❌ por a trela (wrong verb).

Is falou que acceptable in Portugal for “said that”?

In European Portuguese, prefer disse que.

EP: A veterinária disse que…
BP (Brazil): A veterinária falou que… is common there but sounds nonstandard in Portugal. In EP, falou de/sobre means “talked about.”

Any quick pronunciation tips (European Portuguese)?

Approximate EP pronunciations:

veterinária: veh-teh-ree-NAH-rya
gata: GAH-tuh
deve: DEH-veh
trela: TREH-luh
sala de espera: SAH-luh d(ɨ)sh-PEH-ruh

Stress falls on the capitalized syllables above.