During Neelix's farewell to LT Paris he says his name is Tom J Paris but it's been stated several times his name is Thomas Eugene Paris. - Incorrect. Neelix says "Eugene" and the closed captioning agrees.
Neelix has a small recording device that he turns on for his Briefings program. The device sits on a table or counter where Neelix places it. There is no one else around to control it, and it zooms in and out on its own to focus on one person at a time, or more than one.
There are cameras now that can do this using a form of AI. This could also be done by Neelix using a remote of some sort to control the camera.
If the transporter room is capable of transporting a communications badge out of engineering, then they should also be able to transport a security team into engineering.
Neelix's ability to reuse an overheard command code from an engineering officer to gain access to a restricted computer points to an obvious security flaw. If other Starfleet officers were trusted to overhear these codes, there wouldn't really be a need for codes in the first place; and in some episodes these codes are spoken within the hearing of visitors to the ship or even enemies.
Neelix, asked to continue his investigation by Captain Janeway, returns to Engineering where he asks to once again see the communication logs. Lieutenant Torres asks Lieutenant Hogan to help Neelix retrieve the logs. From there, Neelix and Hogan discover that someone has been sending unauthorized communications through the ship's power grid via the waste energy, yet Hogan sees no reason to notify his superiors of the new discovery.