Learn Spanish (Latin America): ¿El paciente se encontraba muy cansado en el quirófano? - Was the patient feeling very tired in the operating room?

This is a great example of a verb changing its meaning when made pronominal. The base verb encontrar means "to find". But the pronominal form encontrarse (using the pronoun se) changes the meaning to "to feel" or "to be in a certain state or place" (literally, to find oneself). You could use se sentía or estaba here, but se encontraba is a very natural, slightly more formal alternative.

Why is the verb in the imperfect tense (se encontraba) rather than the preterite (se encontró)?

The imperfect tense is used here because it describes an ongoing physical or mental state in the past. The sentence is asking about a continuous condition (being tired) without a specific start or end point. If you used the preterite (se encontró), it would sound like a sudden event, as if the patient suddenly became tired at one specific moment.

Is paciente used for both a medical patient and the adjective for having patience?

Yes! Just like in English, paciente acts as both a noun (a person receiving medical care) and an adjective (being willing to wait). Here, the article el tells you it is being used as a noun. If it were a female patient, it would simply be la paciente.

The word quirófano doesn't look anything like "operating room". What is its background?

Quirófano is the standard medical term in Spanish for an operating room. It comes from Greek roots meaning "hand" (as in chiropractor) and "to show". It originally referred to an operating room with a viewing glass so medical students could watch the surgeon's hands at work.