Does rosacea cause facial swelling, burning or itching?

Yes. Severe rosacea can cause swelling, itching and a burning sensation and would require oral antibiotics and perhaps even low doses of steroids to calm down the condition in tally before the topial medications can work.
Sometimes. Rosacea is associated with heat. This can be temperature from the sun, hot foods or drinks, and/or spicy foods like peppers. When symptomatic, it may be associated with sweating, red flush to the skin, and possibly swelling, but the symptoms do subside after a time.