Continuity mistake: During a conversation with Kramer, Jerry is holding a drink in his left hand. Although he is never seen setting it down, the drink disappears in the next shot and he has his arms crossed. (00:02:10)
Audio problem: The end of the word "bluey" that George is saying is cut off suddenly at a change to the shot of Jerry. (00:20:20)
Plot hole: When Elaine meets Glenn's welfare case worker at The Apartment, the case worker does not tell Elaine that Glenn actually has a wife (after clarifying that she isn't The Wife). This is pure plot convenience. When Glenn's wife finds Elaine in The Apartment, he says "You're home early." She lives there, they are not separated, and surely the case worker would be involved with both of them, not just Glenn.
The Calzone - S7-E20
Other mistake: When Kramer is at the restaurant, as a favor for George, he tells the employee that he wants "three calzones." Then the employee just starts cooking the calzones. He never asks what ingredients/filling Kramer wants in those, and Kramer never says.
Other mistake: While Jerry is looking out the window, to his right, there is a Yankees baseball cap on top of his mini basketball hoop, above his computer monitor. Jerry in real life is a Mets fan, and he alluded to being a Mets fan on the show as well (taping a Mets game, wearing or showing Mets paraphernalia). In no universe would a Mets fan have a Yankees cap on display and vice versa. This was before George got the job at the Yankees so that can't explain it. Was it just bad set design and props? (00:14:22 - 00:15:30)
Suggested correction: There's a Yankee hat in many previous episodes. It's never mentioned where it came from, but it could be a gift or signed or hold significant value to Jerry as opposed to him buying himself a Yankees hat to put on display. And that's assuming Jerry in the show is such a Mets fan that he wouldn't like the Yankees as well.
Factual error: Elaine is placed in the same jail cell as the others. No police department anywhere in the U.S. would put a woman in the same cell as men.
The Suicide - S3-E15
Other mistake: While George is talking to Elaine and Jerry in Jerry's apartment, Jerry and George talk about how George's mother and father are both bald. We know from later episodes after his parents are introduced that they both have hair.
The Old Man - S4-E18
Revealing mistake: After The Old Man's dentures are destroyed in the garbage disposal, he shouts that his dentist is "downtown." There is sibilance audible in his production of the T sound (after the stop release). Sibilance is high frequency noise cause by turbulent airflow directed at the front teeth vibrating randomly in a small space; if he had no front teeth, there should be no sibilance. (00:14:35)
Revealing mistake: When Kramer is driving his Tupperware full of blood in Jerry's car, you can tell by how rapidly The Blood is oscillating that it is just red-colored water. As the saying goes, blood is thicker than water, so actual blood wouldn't oscillate that rapidly, especially when it had just been pulled from a freezer. (00:18:20)
Other mistake: The previous episode "The Deal", was supposed to be the season (and possibly series) finale, however it was aired out of order. Therefore in this episode and the next without explanation Jerry and Elaine are not romantically involved anymore, and Kramer pitches cable to Jerry as if he never had it before, mentioning "The naked channel" that was sorta the triggering factor of "The Deal" 's plot.
Suggested correction: As you said, the episodes were aired out of order. That doesn't make it a mistake because if you watch the episode in the order they were produced, there would be no mistake.
Other mistake: When George is taking off his sweater revealing the chocolate-smeared shirt, a woman in the background hilariously stares, looks down at the shirt and then eyerolls. She is behind him, though, and can't possibly have seen the stain. The shirt looks fine from the other angles. (00:16:30)
Continuity mistake: Kramer tries to stop Jerry from running towards the door. "Agent Stone" in the following shot has his badge in the left hand (was in the right earlier) and reaches with the right for his gun under The Jacket. But there's another shot, with cornflakes and other things being blown up by bullets, when Stone is still showing his ID in the right hand. (00:09:00)
Continuity mistake: In the shot when she protests that the husband with the chipper name is only a Kennedy by marriage, Elaine is holding a paper towel, not quite in continuity with what comes before and after. (00:01:35)
Continuity mistake: During the visit, whenever George asks questions about hot water (can I use it on my face, can it be lukewarm) his hands have a sudden position change between shots. (00:14:50 - 00:16:00)
The Revenge - S2-E7
Continuity mistake: During Elaine's amazingly off-putting flirting with George's boss, she starts mentioning how people don't say "Bless you" as they used to. She is gesturing with her right hand pointing a thumb -but before her "Have you noticed that?" is over, she is already in a new shot with hands joined and fingers entwined. (00:16:00)
Continuity mistake: Jerry gives Tor the piece of paper to decipher; the paper is curled towards him in the reverse shots, and Tor's hand is placed at a different height - just look at the fold. (00:17:30)
The Revenge - S2-E7
Continuity mistake: George walked back into a meeting at his workplace pretending nothing happened. When his boss enters the room, George repeats in two separate the shots, differently, the gesture to hide his face behind a folder. The folder itself changed position before the boss arrived, since it was in his hand at first, then on the briefcase. (00:08:00)
Continuity mistake: The belligerent paramedic shouts at the driver "I want that Chuckle back, do you hear me?" pointing a finger at the opening, but his arm is on his own leg in the following shot. (00:18:10)
Continuity mistake: George grabs the slip of paper from Elaine's hand. When he has his 'heart attack', in the shot when Jerry says "What does that mean?" the paper goes from his left hand to his right hand. (00:02:45)
Continuity mistake: Elaine hands the ice cream bucket to George. George is holding the plastic spoon against the container, in his left hand. In the close-up he reaches for the spoon off-camera with his right hand. (00:20:45)