1. It is never said that the call was made from Kathy's house. It would be foolish on the part of Vaughn, knowing the call could be traced there. He could have easily made it to a phone booth somewhere else and made the call, or had an accomplice do so.
2. Kathy's house was in a secluded area. There were no other houses nestling next to it. It had a lot of land and trees around it. The nearest neighbor was nowhere to be seen. Even if a neighbor was close by, it would have still taken a few minutes to get there, long enough for Vaughn to get to another phone and make the call. Even if a neighbor was close by, it's not as though going to a neighbor's house would have made it impossible for Vaughn to make the call.
3. The notion that Vaughn was not wearing gloves comes from Steve's dream sequence where we see the killer as Steve pictures him. It was never said that any strange prints were found in Kathy's house. If that was the case, Steve would have been exonerated in the second scene. He is only picturing someone without gloves, and though that goes against the evidence that exists, and the knowledge that a professional would have used gloves. It is what we are seeing through someone's eyes – images their brain is providing them. When we imagine things, our brains provide us with imagery that goes against what we know. It is just the brain instantly providing an image to work with. The director could have had Vaughn wear gloves in the dream sequence, but even though he wasn't wearing them, he could have wiped all surfaces he touched clean.
4. GSR tests came out in 1974, so there were around for his storyline. In Perry Mason's time, they used paraffin tests, which tested for nitrate, a substance found anywhere in the environment, like in soil pesticides. We're thinking too much of what the forensics would say, and not what the prosecution will say to poke holes in it. Prosecutors will try to find a way to explain a lack of or presence of evidence. They can say Steve washed the GSR over his hands, and wiped it off his jacket, or wore gloves and disposed of the gloves on his way to or from the phone booth. The amount of gunshot residue on his hands would also depend on how far away Kathy was from him when he allegedly shot her. If the victim is shot from several feet away, the gunpowder particles will disperse more widely, having more air to travel through, and not a large amount lands on the shooter's hands. GSR has the consistency of flour and can easily be wiped off. Also, even though I agree that the lack of GSR makes the idea of Steve being the killer questionable, the other evidence against him – his gun being the murder weapon, the anonymous phone call describing him, and his story about the phone and the radio not working sounding made-up – are enough to give the prosecution reason to go after him. Even Danno mentions a lack of a motive, but the DA says a grand jury won't care when they see all the other evidence.
5. I also wondered why no one is concerned about the second man, but Steve's men and Che are. The DA is on Steve's side, but he has to go with facts. As of that moment, they found no physical evidence that another killer was there. There is Steve's concussion and the drug in his system. The prosecution could argue that Steve hit his head during a fight with Kathy and got the concussion then. A poster on the IMDb suggested that the prosecution's excuse for the drug in Steve's system was that he injected himself with it to make it look like someone else did, and threw the syringe and the bottle away. It's plausible, but clutching at straws on the prosecution's side.
6. Steve was not in charge of his own investigation. The press made it look like he was. Danno was in charge. Manicote even yelled at Steve for being involved in his own investigation, asking him if he wanted to lose his bail. That shows Steve was not officially allowed to be involved in his own in investigation. All he did was question his tennis instructor, who confessed that he was forced to do steal Steve's gun. If the instructor's sister was not at the hideout like he said, Steve could have gone to jail just for violating his bail.
There is a lot more of value in this episode than just Camilla Sparv. Getting a glimpse into Steve McGarrett's private life, and seeing him so distraught and upset over Kathy's death allowed Jack Lord to show more of his talents, and made the storyline more impactive. The chase scene on the beach was killer, and Danno, Chin, Duke, and Che all got to their parts to clear their boss. Danno's tense scenes with the DA shows he's a man who can hold his own and is fervent in his quest to fin the real killer and exonerate Steve.