Very important question, surprised it hasn’t been asked. Yes they do, the gag reflex is very strong. Source, wife is a nurse.

IANAD, but my understanding is that minimal sedation leads to better outcomes, so patients may be partially awake some of the time, varying the dose of propofol and fentanyl. During these low-sedation times they may do breathing trials to try to wean off the ventilator. (They called this "sedation vacation" in the ICU I visited.) Not sure if this is how COVID victims will be treated, though.