Jurisdiction of Texas Rangers in Louisiana? I understand that Texas Rangers are responsible for killing Bonnie&amp; Clyde in North Louisiana. Is it lawful for a Texas Police Agent to engage a citizen in New York City?
Answering the question title, a Texas law enforcement officer can certainly make arrests in Louisiana these days under the right circumstances (I'm not about to look up the laws as of 1934). For starters, Louisiana law grants any person the authority to make an arrest when the person being arrested has committed a felony, whether or not that felony was committed in the presence of the person making the arrest. This is normally a legally risky thing to do (the arrest is illegal unless the person actually committed a felony, while a cop's felony arrest is legal as long as the cop had probable cause), but in this case the pair had been involved in a kidnapping and a robbery in Louisiana. Any person could have made a lawful arrest, and could have used necessary force to effect that arrest. But suppose the gang turned out to be innocent of the Louisiana crimes. In that case, a citizen's arrest would be illegal. But the Texas lawmen weren't at the ambush alone. They were there with the parish sheriff and a deputy, who were Louisiana peace officers with the authority to make an arrest on probable cause. And under Article 219 of the Code of Criminal Procedure, A peace officer making a lawful arrest may call upon as many persons as he considers necessary to aid him in making the arrest. A person thus called upon shall be considered a peace officer for such purposes. Neither of these things depends on the Texas officers' status as Texas officers. There are some arrests which are legal based on that (e.g. hot pursuit), and a Texas officer has some extra powers in Louisiana based on federal law that make an arrest easier (e.g. cops in the US can carry concealed firearms nationwide without needing a CCW permit), but under normal circumstances a Texas police officer has no special authority to make an arrest in Louisiana. However, it's not at all uncommon for police agencies in different states (or at the state and federal level) to cooperate on something, and there are ways to make it work out. With more planning, there are normally formal ways to do it instead of needing to rely on "we'll ask you for assistance" (for instance, officers could formally be appointed as deputies in the appropriate agency; this happens a lot on federal task forces, where a deputized state or local cop gets nationwide jurisdiction). If Bonnie and Clyde existed these days but the feds wanted to involve state cops, they'd just set up a federal task force, make Hamer a special deputy US marshal, and go from there.
Firstly, there is no jurisdiction in the US where rape is a potentially capital crime, and murder is not - so you are discussing a hypothetical (and rather implausible) jurisdiction. Given your jurisdiction is hypothetical, you can make the law be what you would like. Secondly, duress is accepted in almost all jurisdictions as a defence to a charge of anything except homicide (and some jurisdictions allow it to reduce the severity of a homicide charge). Courts can be reluctant to accept a charge of duress, but a bullet wound would probably be persuasive. I don't know if someone who had just been shot would be physically able to perform though. The precise crimes the perpetrator is guilty of would depend on your specific jurisdiction, but I would expect at least: rape (because they organized it) kidnap shooting the brother
Apart from hiring an attorney (who will, based on the specifics that you tell him, have a better recommendation), you can appeal to the district attorney, the mayor, a higher officer in the chain of command with the local police, and whatever TV-on-your-side news-guys there are. The decision to prosecute rests with the DA: the police make recommendations. The DA probably has investigators, and all of the above can persuade the police to take a closer look at the case. Ultimately, you cannot force the police to investigate, only the local government that the police are a part of can order the police. An attempt to sue the police for not doing what you think they should do will fail, because the courts have repeated held that barring an illegal basis for non-investigation, investigating purported crimes is discretionary.
Does the party have any legal leverage to engage the police (or other competent authorities apart from private investigators) to help locate the witness and serve the summons on them? Not really. Legal process is not infrequently served by a sheriff's deputy. But the deputy will not generally take any initiative to locate a person to be served beyond what it provided by the litigant. The main reasons to have a sheriff's deputy serve someone with process is the fear that the person served might react violently. Or is it just the party's bad luck that the witness cannot be located and served on? Pretty much.
In the USA, you must be found guilty "beyond reasonable doubt". As you describe it, I'd say there is an unreasonable suspicion of guilt, not guilt beyond reasonable doubt. If the magician killed three people that way, then three unexplainable deaths following three spells might get him convicted. A jury might say that even though there is no way to explain how the killing worked, the correlation might be enough to prove guilt beyond reasonable doubt.
Does said police department have any obligation to ID, investigate and detain / arrest the false caller? No. See Castle Rock v. Gonzales, 545 US 748 (2005). Usually, police do investigate, but that is a matter of department policy and political expectations, not a legal obligation to do so.
The powers given to law enforcement professionals will be detailed in the relevant law that establishes them. I would suspect that the decision to cordon off an area would fall within the purview of the officer on the scene; the idea that a police officer would need to seek permission before cordoning off a motor vehicle accident or chemical spill is unworkable. I would also suspect that other emergency personnel (e.g. ambulance and fire-fighters) would have similar powers. However, such cordoning off would be a temporary measure and if it was maintained for an unreasonable period it would be open to challenge through an administrative or judicial process. If the police decided that a feature was a permanent hazard then they could seek a court order on the owner of the property to provide some measure to adequately protect the public, by either removing the hazard or providing some permanent barrier, under whatever laws seemed most appropriate.
It's called police and prosecutorial discretion to discern when to arrest and prosecute; and that situation in particular is also the result of a decision of the jury of the court of public opinion. Permits are required to sell on the street in Oakland. But not everyone who sells has a permit, and not everyone who is confronted about not having a permit is arrested and prosecuted. There are simply too many potential cases to prosecute. And, the police officer has the discretion to ticket or not. When you get pulled over while driving or riding a bike, you don't always get a ticket, since the officer has the option of discretion. When the officer responded and found an eight year-old selling water, he obviously was aware of the fact that it was a violation. But he was also aware of the court of public opinion. What is it going to look like if he arrests an eight year old and their parent? Allison Ettel was right, in a purely legal sense, to make the report. And technically, the child (and adult) needed a permit. And could have been ticketed and prosecuted. But it was Ettel was tried and convicted in the court of public opinion, and she lost her case. Happens a lot.