We learn in Goblet of Fire that there is a spell that can detect what the last spell a wand cast was. They’re not gonna do that to his wand? The only wand in the vicinity, just as a precaution? Muggle cops do gunshot residue tests on every suspect in a shooting, as a matter of course. Are the best dark wizard catchers in the country really less thorough than Barney Fife? I think a lot of Harry Potter can be chalked up to “worldbuilding as the plot requires,” but in-universe you could probably just say that the Ministry is clearly wildly incompetent and corrupt. And that’s why they’re not just useless, but actively harmful throughout the series.
Myrtle was killed by the basilisk. As for the rest I know it’s not a perfect explanation but I think the most logical argument is that we’re talking about Voldemort here. He’s only rivaled by Dumbledore as the smartest and most powerful wizard who ever attended the school. Since the trace is just a standard charm cast by the ministry it’s totally believable that a motivated Tom Riddle would figure out a way to circumvent it. Voldemort framed his uncle for his father’s and paternal grandparent’s murders (his uncle was the only wizard in the area and he had a rapport for muggle-baiting before, and he used his uncle’s wand for the murders not his own.)