the existence of a death penalty NOW is often a major issue when dealing with other countries and things like extradition. Are you seriously suggesting that it will become LESS of an issue in the far future? Asking about a death penalty would be one of the first things Yar asked about, at which point she would have discovered that it was the ONLY penalty for crimes there. That's hardly a meaningless technicality of the law.
The starting point would appear to be that if one of the heroes misbehaves in a way that calls for death by local rules, he dies - that's why there's a dilemma here, because Picard feels local laws must
be obeyed rather than be weaseled out of. So the nature of punishment would be rather secondary, as the visitors would not need to know anything beyond what they can do and what they cannot. If they do something they are not allowed to do, despite having been provided an info sheet by Yar, they are on their own and Starfleet farts in their general direction. (In this episode at least, and until our heroes decide the local laws are so silly and foreign that they can be dismissed.)
one research local customs and laws? In the usual case of a world Starfleet is allowed to visit, there would no doubt be a handshake package of some sort, provided by the locals. And that would read how the Edo want it to read. In which case, see below:
At any rate, even if she didn't study punishments, which is just a silly notion, they still didn't seem to know about penalty zones.
But the whole point of the system is that the offender is not informed about the zones!
It's an excellent way to deal with rampant crime on a global scale with minimal resources. The government might only be able to afford twenty policemen per continent - but even with fairly conventional means of transport, they could be at surprising and unexpected locations in sufficient numbers to subdue any wrongdoer, especially when they are cleared to use lethal force immediately and without any sort of hesitation (say, investigating the nature of the crime, establishing the degree of guilt, or other time-consuming, officer-endangering irrelevancies). Sure, 99.9999999% of the time, they would not be there. But that's why the deterrent needs to be based on horrible immediate death rather than something the wrongdoer might consider a price worth paying.
After a few years of setting examples, the system can be toned down and, say, the "horrible" aspect dropped from the executions...
If the Edo wanted their guests to behave, they would naturally keep the existence of the zones secret from them; indeed, their law would be likely to require them to do so.