Learn Russian: Врач должен дать снотворное, если у пациента сильный стресс. - The doctor must give a sleeping pill if the patient has severe stress.

Снотворное is an adjective acting as a noun. It is neuter because it implies an omitted neuter noun like лекарство (medicine) or средство (remedy). You have seen this before with words like мороженое (ice cream).

Why do we use the perfective дать instead of the imperfective давать?

The perfective дать is used because the sentence refers to a specific, one-time action of giving a pill to a patient to solve a current problem. If it meant a general, recurring medical rule (e.g., doctors in general must give it), the imperfective давать would be used.

Since this lesson is about impersonal involuntary states, how could we say 'if the patient can't sleep' using that grammar?

You would use the Dative case for the patient and a reflexive imperfective verb: если пациенту не спится. This clearly expresses that the lack of sleep is happening to the patient against their will.

Why is у пациента in the Genitive case instead of the Dative case we use for involuntary states?

The phrase у пациента expresses possession (the patient has stress) using the standard у + Genitive construction. The Dative case is used for the experiencer of an involuntary action (like пациенту не спится), not for showing who currently possesses a condition like стресс.