Ben handcuffed Tammy before jumping out of the window into the garbage container. Right when they land, the cuffs come off. When they climb out of the container, the cuffs are on again.
When Jake closes the door of Tammy's apartment and then blocks it with the chair, he doesn't turn the lock. When the men with guns arrive, they rattle the door as if it is locked, and when they kick the door in, the frame breaks near the lock as if it was engaged.
This is set in July 1981. The Human League song featured was on the album Dare which was released in October 1981 and the single wasn't released until November 1981 in the UK (taking the prestigious title of Christmas number 1 that year). It didn't top the charts in USA until July 1982.
When Ben/Eva and Jake flee after Tammy shot Zach, they drive away in a street perpendicular to the one where the henchman is shooting from. And yet, the car is shot in the rear window and the trunk.
Addison instructs Ben to shoot the case that Tammy Jean is holding with the taser in order to shock her, reasoning that the case has metal in it and will therefore be conductive. The handle of the suitcase, the only part touching Tammy Jean's body, is plastic, which is not conductive. The plan should not have worked, but somehow did.
On Quantum Leap (1989), people whom Sam leaped into, like Lee Harvey Oswald and Dr. Ruth Westheimer, were shown to remain conscious while in the Waiting Room. Magic says here that he blacked out when he was there. But it also was established that the leapee has "Swiss cheese memory" from the time of the leap, which easily could be interpreted as blacking out. Also, it should go without saying that not everyone's experience would be the same.
At the apartment building, a safety fence had been built around the pool. California didn't require pool safety barriers until 1997. However, just because something is not required does not mean that it cannot be implemented, especially a safety feature.
At the start, when the bartender pours Ben a drink, the reflection in the mirror is of Ben, not the woman he leaped into.
(at around 16 min) In the action shot of Jake (Justin Hartley) jumping into the dumpster, it is clear that it was a crash bag he jumped onto instead of garbage bags.
Ben leaps to 1981, but an establishing street shot shows a billboard advertising "LA's Classic Rock KLSX 97.1" which didn't exist until September 26, 1986.
A Taser is featured prominently in the story with Ben carrying one as a bounty hunter and using it on another character. Tasers did not become standard equipment by law enforcement agencies, and were not even well know by the general public until the late 1990s.
While in Tammy Jean's apartment, Ben looks at of a pile of modern Mexican pesos, including polymer banknotes and bimetallic coins. The episode takes place in the early 1980s. Bimetallic pesos were first minted in the 1990s, and polymer banknotes a decade later.
The $100 bills in the suitcase are from the 1990s and later.
Much of the fashion indicates more late 1980s/early 1990s Los Angeles, not 1981.