When Darhk is holding Ray's little suit, the grip keeps changing from his finger on the head when showing Ray, then gripping only the sides.
In this episode, Eva suddenly started looking much older than she earlier was looking, especially from the scene when Bennet the killed, thus she ended up being the Director of Time Bureau. Could have been makeup error, or could be intentional on part of the director that a lesbian love bitten young girl would not suit as the Director of Time Bureau.
When Nora and Ray had gone to bring the scientist from 1962, Damien Darhk says "they should be back an hour ago". Though it is not any rule, it is a protocol that whoever goes on time travel, return almost at the same moment he left at originally. This removes the gap in his knowledge that might come due to whatever might have happened in the gap of the duration he was gone to past or future. So, They didn't come back in an instant should have rather warned Darhk that something crucial had gone wrong (like time travel device broken, in this case), and he should have gone himself to correct things, instead of blabbering idiotic things.
When Nora is falling, Dahrk pulls her up, but doesn't simply set her down on the roof, he keeps her hanging. In a similarly stupid move, his younger self does not shoot him while they are fighting hand to hand with Nora hanging. Although older Dahrk stopped the initial bullets he was then shown as being vulnerable when they were fighting.
If cold fusion had been invented in 1962, it is impossible for it to remain unknown till now. No scientist works alone in total seclusion on such a big thing, he needs inputs, ideas, instruments, funds, and his colleagues get to know at least something about it. And here, he is even guarded by dozens of security personnel, so, even if one scientist gets killed, the word of mouth goes around that cold fusion has been invented, leading others to work more and more on it seeing its gigantic potential, so someone else would have had invented it soon again.
There were so many security personnel guarding the scientist in 1962, but nobody picked up the pieces of Time-travel device that got broken when young Darhk had fired on Nora. Even if young Darhk had killed all security personnel present there, backup should have arrived as they must have some alarm system in place there to alert about the attack. Elder Darhk comes and picks the pieces himself. Nora, or even Ray who was a scientist knowing this well, shouldn't have left the pieces lying there because she knew that this being a scientific research facility, the pieces would eventually reach the hands of someone who would be able to figure out what it exactly was and would discover time travel before due time, thus creating a massive anachronism, having consequences.
Late Dr Martin had invented tiny device that would communicate across the time line. It should have become instantly popular among all time travelers, and every one must have kept this device while going on time travel. Why didn't Nora-Ray took such a device with them when they went to 1962, so that they could have communicated with Darhk when the time travel device was broken.
Ray unnecessarily and, who knows, intentionally, wasted time when he goes to save the scientist, and elder Darhk could shot the scientist. As soon as Ray had reached there, he should have taken the scientist away to safety, instead of explaining and talking so much. All the seasons are full with such irresponsibly and shabby behavior of the Legends leading to destruction and death. Rip and the bureau and Eva were right in calling the legends idiots and sacking them and keeping them away from anything.
When young Darhk thrown Nora from the roof top and she is falling down to the road for sure death, elder Darhk uses his magic to stop the fall of Nora and lifts her back to roof top. He should rather have made the Nora land on the road safely so that he is beyond the reach of young Darhk gun, while young Darhk wonders what elder Darhk is doing and how. By lifting Nora up, elder Darhk makes her go up and down several times while the two Darhks fight, this might have given Nora a heart attack from fear of falling to death.