It's eventually established that Rya's isotope injections allow Dr. Rao to remotely kill people from the future. But why is this complicated technology necessary? Why didn't she simply shoot them?
Also, wouldn't it have made more sense to kill them in a less-dramatic way, which attracted less attention? Wouldn't the strange hemorrhaging make authorities investigate the deaths a lot more closely?
It's possible that Lock knew the rest of the department would write off the strange method of death. As for the weapon, it's possible that this was the only way to make sure they could be killed without affecting the time stream.
There's a possibility that the original plan was to carry out the killings *without* it looking like the work of a serial killer. A few people would have hemmoraged mysteriously in 1988, then in 1997, then in 2006...but it would just have been written off as a bunch of isolated cases. The problem is that in 1988, Rya botched the killing of her last victim - the girl at the nightclub, by being seen by her victim. This led to the launching of a city-wide manhunt against her and the cops eventually confronting her - making it clear that the killings were done by a serial-killer. Of course, by that point, Rya was at the end of her mission, from her chronological perspective, so there was no way to change her method.
Because a harmless (until the isotope is triggered) injection is quicker and less conspicuous: you approach the target, ping it and move on without them even noticing. Consider that, out of all her victims, only one has noticed something was being done to her, while everyone else just moved on with their lives and died without suspecting a thing (and that includes those killed after the first or second wave). Guns, on the other hand, are messy, noisy and attract attention, so it would have painted a target on her back and made it harder for her to reach the others. Yes, in the long run the mysterious isotope attracts more attention but allows her more freedom in the short run, which is what she needs to move between her targets and inject them. Not to mention that a gun could always miss or result in a non-fatal wound, however slim the chance, while destroying the brain is 100% effective.
At the end of the movie, in 2015, Thomas Lockart reveals to Rya that he killed her in 1988. Rya answers that "it already happened ; if it begins with you warning me here in this beach then it always ends with spoiler: me dying." So, according to her, she can't avoid death because it already happened, implying that it's impossible to change the past. But the whole point of her mission is to change the past!
The first thing to consider is that her main goal is to convince Lockart (a man shown to be prone to obsession, to put it mildly) to move on with his life, so the main goal is to get him to stop rather than be accurate. The second, she is changing the past by moving back in time with knowledge of future events, while Lockart is moving normally through time, so what she is telling is that Lockart can't change what he already did. She could theoretically try to save herself if she knew how she died, but she doesn't seem to care that much (whether because she is more focused on Lockart or she has no interest in her own survival, or a mix of the two is up in the air). An interpretation of her actions in 1988 is that she actually tried to neutralized Lockart and believed to have avoided her fate, only to be surprised by the taser and fall to her death because of it.
Why is it that everybody talks about Holt as if he is some kind of jerk? Even he himself confirms that. And yet none of his actions in the movie are unreasonable or unkind. The worst, perhaps, is when he reveals the existence of the "copycat" to the press; but his reasoning that a hint from the general public might be helpful is not unsound. And him trying to stop Lock investigating things in 1988 is just what you'd expect from a superior officer when a uniformed cop starts playing detective for no reason. Everything else is just... kindness.
