Near the end, when Chandler hugs Rachel in the hallway after she was talking about the 4th of July, the door to Chandler and Joey's apartment is open. But a second later, Ross opens the door as he comes into the hallway.
At the end of the evening, Joey, Ross and Chandler all dance in the hallway to distract from Rachel's father going into the apartment. When the door shuts behind Doctor Green, Rachel is in the middle of the group hug and her mother is on the left. In the next shot, Rachel's mother is in the middle and Rachel is on the left.
At the end, Joey kisses Rachel's mum to distract her, as Rachel leads her dad down the hallway and presumably down the stairs, yet in the very next shot she is standing next to Ross, right behind Joey and her mum who are still kissing.
When Ross goes over to Monica's apartment to get Dr. Green's cigarettes and glasses, Mrs. Green tells Ross that she didn't know he smoked. Ross puts a cigarette in his mouth to show that he does smoke. The cigarette is in his mouth backwards, but when he goes back out into the hall and sees Dr. Green, the cigarette is in his mouth the correct way.
Ross asks Rachel's dad about how things are in "the vascular surgery game". Later in the series (in the one where Joey speaks French, season 10) it is said he is a cardiologist, as it was ironic he had a heart attack.
The gag when Ross is going out of Monica's apartment (wearing Rachel's dad glasses and with the cigarette on his mouth) was that he stands still, but by an editing error it's clearly seen than Ross retakes position and then stand still.
When Ross meets Dr Green in the hallway after fetching him a scotch from Monica and Rachel's apartment, he proceeds to stop Dr Green from entering the apartment to retrieve his glasses and cigarettes by saying to him "you can't go in there mister as that is the Staging Area". How could Ross have possibly known to refer to it as this. He couldn't have, as he hadn't been in the apartment in a earlier scene when Monica, Joey, Chandler and Phoebe were present in the apartment with Dr. Green when Monica said to Chandler and Joey that "This isn't where the party is being held dummies, This is just the staging area". Ross and Rachel only appeared in a later scene.