Upon meeting the Zimmerman hologram, the Doctor comments that he now sees where he gets his hairline from. The Doctor had already seen what the real Zimmerman looks like in the season two story "Projections."
Janeway suggest that Kes keep the Doctor's mind active, "force him to use his neural matrix," as a possible way to slow down his degradation. Continuing to use a damaged computer component and taxing it more heavily than usual is a good way to ensure that it breaks down more quickly, not to preserve it.
The Doctor somehow remembers Jupiter Station, where his program was created. It was previously established that he was not brought online until he was installed on Voyager and has no memories prior to being activated onboard. While he may have an awareness of Jupiter Station, he would not have memories of it.
No explanation is offered as to how Voyager's scans of the damaged freighter managed to miss the Swarm ship very noticeably stuck to its hull.
Janeway orders the shield harmonics modulated to protect from blow back from firing the phasers at the Swarm ships that are attached to Voyager's hull. The shields extend outward some distance from the hull, forming a bubble around the ship. The Swarm ships attached to the hull would be within the bubble and the shields would not protect from damage they might inflict. No mention is made of adjusting the shield diameter.
When getting ready for Paris' surgery, The Doctor states that he is sterilizing his hands, which makes little sense: he is a hologram that could just change his density to nothing, remove any germs on his surface, and then reinstate his solidness. Furthermore, after the sterilization is complete, he touches the control pads on different pieces of equipment, which have been touched in the past by other people who were not sterilized and would nullify the effort of sterilizing his hands.
Janeway angrily says she will not add 15 months (or more) to the crew's journey by going around the Swarm's space, yet she is constantly adding time throughout the series by not traveling at warp and stopping to explore random phenomena. The ship was just earlier not moving at all while Paris and Torres spent 5+ hours on a survey mission.
Janeway orders Tuvok to sweep the pursuing Swarm ships with the forward phaser array, which he does. As the ships are all behind Voyager, it would make no sense to fire on them with the forward array. The ship has aft phaser arrays for a reason.
Janeway orders Kim to modulate the shield harmonics to protect from blow back when firing at the Swarm ships. The shields are controlled by the tactical officer, in this case Tuvok, not the operations officer.