Other mistake: A bottle of prescription pills dropped onto a cement parking garage floor is not likely to have its cap come off (unless it was already loose). If the cap does come off, the pills (especially round ones) are not likely to settle around the bottle, but the police photo showed the pills clustered around and in close proximity to the bottle. Also, Alex - a professional assassin - did not bother to at least pick up the prescription bottle that had his name and other identifying information on it. (00:44:30 - 00:48:30)
Other mistake: When Alex was trying to cauterize his shot wound with a lighter after he poured drinking alcohol on it, the fire tightly formed around his bullet hole but it would have engulfed his entire pant front as well.
Other mistake: Alex's prescription bottle on the table appeared to have 4, possibly 5, pills in it. When he heard the FBI agents coming into the bakery, he held the bottle up to his mouth and took an unknown quantity (apparently, more than one) then threw the bottle on the floor. Vincent brought the prescription bottle to Alex in the hospital. It could not be seen how many Alex took (again, apparently more than one). The rattling sound afterwards suggested there were still several pills in the bottle. (01:16:58 - 01:27:41)
Other mistake: Despite all the blood that dripped from Alex's right hand and amount of blood on his stomach/shirt from getting shot, after he killed the motorcycle patrolman it could be seen that no blood had dripped onto Alex's white shoes. (01:10:07 - 01:12:40)