The brand name of the dough presser is obscured by flour in the close-up, but is clear in all other shots.
Ricky is just about to give Lucy a check. In the next shot, Lucy has the check in both hands and Ricky is fixing his tie.
When Lucy is at the pizzeria attempting to spin the pizza the tray of dough balls in front of her goes from nearly empty to full between shots.
The way the pizza is made in the pizza parlor is factually incorrect for a true New York pizza parlor. When Lucy makes a pizza she takes a ball of dough and feeds it through a machine to flatten it out before tossing it. This was the way pizza parlors in Los Angeles made pizza at the time. In contrast most New York pizza makers would use a rolling pin to flatten out the dough before tossing it. This allows the maker to control the thickness of the crust as they see fit.
Dominic says he is a native Italian, and his Italian family has been living in one of the most famous Italian cities for generations, Vienna. Vienna is the capital city of Austria and, even though the two countries butt up to each other, Vienna is nowhere close to the Italian border.