Would alcohol cause vasodilation in the brain as it causes vasodilation in the body?

Not really. You mean, "Does your brain have that cute little red nose and rosy cheeks" appearance? Although some degree of vasodilation can occur in cerebral vessels...the majority of cerebrovascular dilation or constrictions controlled by the nervous system itself and will not respond to normal levels of alcohol. That is because BP within the confines of the brain are EXTREMELY tightly controlled.