I understand that Texas Rangers are responsible for killing Bonnie& 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.
A Texas Ranger is not a "Texas police agent"... he is a Texas Ranger. As per the charter which created the Texas Rangers, a Ranger's jurisdiction extends to wherever he stands, be it in Austin, Argentina, or Juarez...
Of course, the government of Mexico might have other ideas, but they'd have to take that up with a man who's got more grit, guts, and brains than pretty much anyone you can find, and they had better get him before he gets the man he's after back across the border.
Their historical refusal to acknowledge jurisdictional borders results in one indisputable truth. If you have done something evil enough to get a Texas Ranger on your case, you're screwed. There is nowhere you can run, and nowhere you can hide where he will not hunt you down. You will become that man's life's mission, and you will either face justice or die by his gun trying to avoid it.
Keep in mind, Texas Rangers do not generally handle your run-of-the-mill crimes, they will not show up at your door because they are following up on a stolen carton of cigarettes. They handle crimes such as violent murders and confirmed threats or assaults on people of certain, special groups such as widows, veterans, handicapped persons, and the elderly. So the people they are after are BAD people who would make any law enforcement officer nervous. They have a reputation of keeping their revolvers cocked and maintaining a "shoot first, ask questions later" policy.