Audio problem: Lorraine comes to see Marty in Doc's garage in 1955 to ask him if he will go to the dance with her. Just before Lorraine moves to the other side of the car from where she walked in, she begins to say something to Marty. The problem is that her lips don't move when she talks, I guess the right clip ended up on the editing room floor. (01:11:40)
Audio problem: When Marty's band is auditioning at the beginning of the movie, the drummer's hands are inconsistent with the drumming sounds. Especially obvious when they get cut short and we hear a mini drum roll, which the drummer definitely doesn't play. (00:08:15)
Audio problem: When Marty is playing the intro to "Johnny B. Goode" you can hear some cymbals in the background. But you can see behind him that the drummer isn't hitting any. (01:27:30)
Audio problem: Marty's singing during the "Johnny B. Goode" number is obviously dubbed; it goes out of sync several times. (01:27:40)
Answer: The video camera was in the DeLorean. With the right kind of adapter, which was common enough in the 80s that Doc might've had it on the camera or been able to jury-rig something in the 50s, it would have been possible to connect it into the antenna screws in the back of the TV like an old Atari and play it directly from the camera.
Captain Defenestrator
TVs in the 50s had a two prong antennae connection (two screws in the back that you put a prong antennae into) TVs in the mid 80s also had this. The coax connection (the one wire that screws in) was starting to become common, but, the two prong connection would have been more likely on any given TV at the time, so, whatever wire they used to preview recordings probably had that. very convenient that Marty brought those cords with him.
An old Atari 2600 RF Adapter would be how one would link a video camera to an old-fashioned television. A simple-enough part that Doc could probably make one with 1950s technology.
Captain Defenestrator