When Phoebe tells Mike she has changed her name, Mike's scarf alternates between being tied and not being tied three or four times. Later, when they talk to Rita, Phoebe's client, the same happens.
After Rachel tells Ross she was fired, Ross hugs her and puts his face against hers twice.
Phoebe would not be able to change her name in a single afternoon. In New York City, the process takes a couple of months and involves, among other things, publishing the name change in a legal periodical.
Ross says since he got tenure he cannot be fired and has job security for life. This is highly exaggerated and incorrect because when an instructor receives tenure, for whatever reason they maybe fired tenure entitles them to due process and can plead their case.
Generally speaking, most labor laws prohibit an employer from terminating an employee for looking for a new job.
When Joey is in the new house sitting in the playroom, he is holding a flashcard that says "12+6." When he flips it over and makes a noise of frustration to imply he got the problem wrong, the viewer can see that there isn't an answer on the other side, just another math problem. However, many flash cards are two-sided with a solution in the corner of one side to the problem on the other side. The answer to Joey's 12X6 problem was likely in the upper corner on the other side of the card.
Visible in the first scene as they all sit down to dinner.
Phoebe expresses shock when she discovers that anyone can legally change their name to anything, for any reason. This contradicts The One with Rachel's New Dress (1998), where Chandler tricks Phoebe into naming one of her babies after him by threatening to "go to the courthouse" to change his name.