When Dr. Meltzer is breaking into Angel's apartment, one hand is rattling the grate while the other opens the door to let him in. However, when he enters, he lifts his hand to help guide back into place his disembodied eye, seemingly while both hands are still detached.
In the opening scene the invoice Cordelia is holding switches hands between shots.
After Ronald enters Melissa's new pin into the ATM machine he doesn't specify the amount of money that needs to be taken out, yet money still comes out of the machine.
Ronald should not have been able to change Melissa's pin number at all. Almost all banks now require the card holder to actually show up in person at the bank with the card in order to change their PIN.
Towards the end of the episode the mad doctor shoots Angel with an animal tranquilizer that slows down and eventually stops the heart. Angel falls to the floor, breaks out in sweat, has tremors, and lets the bad guy get away. As a vampire, Angel's heart doesn't beat so this scene makes no sense.
But, Angel wasn't actually dying. He was pretending to die, so the bad guy would think he got away with it. Once he thought he was free to kill the woman, Angel showed up and surprised him.
When Ronald enters Melissa's new pin number, the sound effect only uses three beeps despite there being 4 digits.
Dr. Meltzer may have perfected some way of being able to detach and reattach his body parts, but it's all based on some medical technology and a little psychic surgery that he's perfected. Nothing in his research would allow his eyes to float around unattached. He's not supposed to be supernatural, just a nasty piece of work with a skill he's refined.