The 90's animated Luthor is more in line with his DC Comics counterpart after the mid 80's, he matches up almost perfectly with Luthor during the Superman is dead comic story and after, smart enough to be a great scientist but dirty enough to be an evil politician.
I think Batman as a super scientist fits, never had an issue with it. He supposedly immersed himself in all things educational for almost 10 years during his young teen-young adult years after his parents were murdered, according to the big book about Batman that details his every appearance, villian, etc.
Also notice that 1/2 the time Batman just uses common sense and reasoning and not necessarily any form of super hard science to figure things out, point in case is this 4 parter, its not like its hard to see something is wrong and hunt it down.
As for the gun I completely agree it was Martian Manhunter, but Batman has not been on the Watchtower in a LONG time, remember he is part time
, I would think Atom would be the only other one who would KNOW how to do it, he does not have that side to him to do actually do it but he understood the nanotech well enough I would think remote controlling the gun would not be too hard lol