The actor portraying Sheldon's father in the VHS tape was in a previous episode portraying one of Leonard's high-school bullies.
In response to Amy's comment on adding observer dependency to their theory: Sheldon proposes they can make a combined symmetric/asymmetric version of their theory consistent to all observers by adding extra dimensions. This would not be needed as any string theory or equivalent would need to be relativistically invariant from the outset, else there would be no point to it.
When Amy holds the VCR tape, you can see that all of the VCR tape is on the left side. She puts it in and it starts playing. The VCR should have rewound the tape to the beginning before stopping, THEN Amy could press "Play".
Despite being a genius, Sheldon did not think to remove the tab on a video cassette he intended to preserve, which would prevent it being recorded over. But even if he had his father could have put some tape over the hole to allow more recording.
When Howard is doing his magic trick he says he doesn't have any brothers or sisters but in an earlier episode he found out he has a half brother from his father. However he is referring to full siblings.
After watching the tape of his father, Sheldon uses a sports metaphor regarding his and Amy's super-asymmetry paper and says it's the first time he's done so. He's actually used such a metaphor at least once before; when he banned Penny from the apartment for various infractions in "The Panty Piñata Polarization (2008)", he used the baseball metaphor of a batter being called out on strikes.
During the Viking funeral, three doctors fail to think ahead enough to open the bathroom window. Also, a plastic shower curtain is not flammable. It would simply melt, not combust.
When Amy is coming in the door with the VCR, she sees that Sheldon has changed the name of Fun with Flags by crossing out the word fun and replacing it with fine. On the board, the word fine is written with a lowercase f. Since it is replacing the first word in the title, it should be capitalized. Sheldon would not make a mistake like that, let alone leave it if he had.