The short version of Dave's answer is the DCU more or less amended the Constitution into oblivion. Personally, I don't think I can subscribe to the theory, not just because of the unlikelihood of that happening, but more importantly, I remain skeptical that such changes would affect the aspects of courtroom procedure I had asked about.
But Dave still makes some very good points about how the real world would respond to superhumans, and alludes to the fact that, while we may not admit it often, modern supervillains are frequently just costumed terrorists. Mere bank heists are old hat; villains who threaten whole populations have a lot more in common with Eric Robert Rudolph than a lowly bank robber.
One particular point I want to address is the notion of 'superhuman crimes,' and the suggestion that they would actually have a different set of rules. I've seen this suggested before, and I have my doubts about it. But even if it were the case, I have further doubts that the Sands case would fall into that category of crime.
The Shadow Thief stabbed Firestorm. He stole the stabbling implement through the use of his powers, but the murderous violent act itself was pretty ordinary. Was it a 'superhuman crime' because the victim happened to be superpowered? Because Sands possessed a super-scientific belt? Shoving a sword through a person's gut is aggravated assault, and even though most people don't blow up as a result, death would be a pretty common reaction to such a wound. Take away the costumes, and you're left with something awfully similar to a suspect stealing a cop's gun and shooting the cop's partner. There's nothing terribly superhuman about it.