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.
Continuity mistake: When the cable guy tells Jerry it'll cost him 400 bucks, he is holding the plate with the cake in his right hand, but in the following shot the left hand is holding just the fork. (00:19:10)
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: George walks confidently to his ex and calls her by name. Leslie's friend is taking a bite from a little celery stick, but in the following shot her hand is lowered in front of her chest. (00:16:45)
Continuity mistake: In the opening shot of The Baby Shower party, Leslie turns towards her friend and starts talking as a ponytailed woman walks in the background. The same happens in the close-up that follows. (00:12:35)
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)

Revealing mistake: In the sequence when he gets shot by the FBI agents for stealing cable, Jerry Seinfeld is wearing ear buds, quite visible when he collapses by the door. Evidently the stage props even loaded with blanks were too loud for his taste. (00:09:00)
Continuity mistake: Kramer keeps pestering Jerry with the offer of the illegal cable. On the table behind Jerry, the position of the remote and the cable keep changing between shot in the first part of the scene. (00:06:00)
Continuity mistake: George sits in Elaine's spot and puts his wallet close to a glass of water and a used tissue. When he gets up in the next shot, the glass is further back on the table, past the small plate and closer to Elaine and Jerry. (00:05:40)
Continuity mistake: Elaine sits by Jerry to ask him permission to hold The Baby Shower at his place. Jerry is holding a morsel of food in his hand, gone at the first cut. (00:04:20)
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: Reminiscing about the incident with the performance artist, George says that she was aiming at him with the chocolate syrup "like she was putting out a fire." Notice Jerry has nothing in hand. Next shot and he's clenching a paper tissue. (00:02:40)
Revealing mistake: It's just a parody/absurd sequence, but it's odd that with over two dozen bullets shot from barely a dozen feet of distance, just a couple entry wounds appear on the body of the runaway Seinfeld. Of course no blood either, but that's a necessity given the type of show. (00:08:55)
Suggested correction: It's a dream sequence. It doesn't have to follow the rules of reality. I frequently have dreams that logically make no sense.
I know, I know, but never been a big fan of giving a free pass to dream sequences for things like continuity, poor stunts etc. If anything, it'd get a pass because it's a comedy and violence and realism are toned down by default.
The very nature of dreams give them a free pass for just about anything. I will have dreams where I'm talking to a certain person or holding a certain object, and in the next moment the person will be someone else or the object will be something else. I have dreams where I am back in high school and the layout of the building will frequently change, or the class I go into will change subjects. If you put that to film, it would be a change in continuity.
What you say is true for dream sequences played specifically with the purpose to give the viewer a sense of disorientation, experience something obviously 'off', a deliberately disjointed and creative scenario that breaks reality. As I said, I am not a fan of being unable to nitpick scenes or even movies who happen all in someone's head for trivial mistakes that are not something as amazingly obvious as the ones you explained. Your examples are something the viewer would notice and would register as deliberate choice and part of the plot, but Seinfeld wearing earbuds or 2 gunshot wounds instead of a dozen are not really something I can put in the same category. If the dream scene is played 'straight', as that one has been, I don't believe we have to just assume that any take can be edited together since continuity is not an issue, props and tricks can be visible or act weird because who knows what can happen in a dream, etc.
You make a fair point (which is also why I didn't submit a correction for your separate entry of Jerry wearing ear protection). However, the basis of this submission is that Jerry only has a couple entry wounds and no bleeding after being shot numerous times. That can just be chalked up to how his mind dreamed the scenario. I don't think a sense of disorientation or something being off needs to be established (especially when the sequence is played for laughs) for viewers to accept details like that can suddenly change within a dream since we all dream and understand that those things happen.
Not necessarily "established" but "with purpose", which can be seen in hindsight. Anything can happen in a dream, but if he imagined to be shot in such a dramatic fashion so many times and die, the fact that he dies with a cheap effect is hardly serving any narrative purpose. Again, I could see why ultimately the mistake could be seen as stating the obvious since "the scene is played for laughs", which was my first caveat posting the scene, the last being the lack of blood for censorship purposes. They didn't thoroughly cover Jerry Seinfeld with squibs and things like that just for a gag - explanation of the 'mistake' rather than justification, but fair. But as far as the dream goes, the point of that dream scene is to do something more 'violent' and unexpected than you'd see in the 'real life' scenes, not tone it down through a marginal detail that has a clear explanation.