Why is Victor Maitland such an idiot? Axel had no reason to suspect him until Maitland had him thrown out of his office, at which point Axel was certain that Maitland was behind it. Why not just say, "Mr. Tandino worked security for one of my warehouses....Yes, I do use bearer bonds; that's fairly typical in the art business....No, I don't know why Mr. Tandino had bearer bonds with him in Detroit....No, I have no idea why someone would want to kill Mr. Tandino....Please let me know if there is anything else I can tell you that would help."
For that matter, why does he bother kidnapping Jenny from the warehouse, while leaving Axel there, alive? And why on earth does he bring her back to his own house, of all places? Why doesn't he just shoot them then and there? Or take them to some remote location, kill them there, and dump the bodies? Why does he leave his thugs there to spend several minutes beating Axel up? What was the point of all that, except to let Billy rescue Axel, and then let the two of them, with Sgt. Taggart, rescue Jenny? Seriously, why is Maitland such an idiot?
Because he's a 1980s action movie villain.
The Hollywood Law example is rather interesting because, instead of ignoring Maitland's fourth amendment rights, Axel is actually needlessly sensitive to them. When he and Jenny go to search Maitland's warehouse for drugs, Billy asks why he can't come along; Axel explains that, because Billy is a cop in that jurisdiction, and they don't have probable cause, the judge would rule the search illegal and toss the results. Of course, realistically, any judge with a triple digit IQ would also hold that Axel was acting as an agent of the Beverly Hills police, so that justification would not fly. The real reason they could legally search the warehouse is that Jenny is a manager in Maitland's company and has a key. If the police are admitted to a place with the consent of a person who would normally have access to that place, then they do not need probable cause. If the manager of a business willingly permits the police to search said business, the police do not need a warrant.
Why in the hell would the Alphabet Bandit robberies require a rare gun such as the .44 Auto Mag? It's pretty clear that if they'd used a gun with more commonplace ammunition and not a combination of actual .44 AMP rounds and chopped down .308s to fit the Auto Mag, then Axel would never have got onto them in the gun club and they may possibly have got away with the whole scam.
That question can be most reasonably explained with the Cool Guns trope. One or more of the goons thought the .44 Auto Mag was a kickass gun, and just had to take it. If they'd used revolvers instead, the entire movie would have been a wash... So, lucky us for dumb mooks.
This is confirmed in-movie as Dent calls the mook in question on the carpet for making such a foolish decision, and later has him killed for it.