At least we did not have to suffer the TOS method of having a Vulcan run around using the magical nerve pinch to save the day.
Yeah, so let's just have our Vulcan science officer act like she's a tramp instead, that's so much better than having her act strong and intelligent or anything like that.
And to be frank, doing more than they DID would have violated established continuity about the Ferengi even more than having them run into and plunder a 22nd century Earth ship in the first place. It might not have made a buttload of sense to let them go without finding out what their race was called or where they came from, but if Archer HAD forced the words "Ferengi" and "Ferenginar" out of them the fan community would have been up in arms far, far worse than it already was.
Which is why they shouldn't have done the Ferengi at all in ENT, because there are more things to consider than just if they know who the Ferengi are before Picard and crew catch up with them.
I don't think Archer & Co. had any authority to arrest them. They were in the middle of deep (possibly unclaimed) space. For example, if an American police officer was in China and witnessed a crime he would not be able to arrest the criminal and take him back to an American court to stand trial. He wouldn't have the authority.
I assume Archer was in the similar situation. He probably couldn't (and shouldn't) arrest citizens of a foreign government. For all he knew the "pirates" weren't breaking any of their own laws.
But I could be wrong.
Yes, you are wrong. This would be more akin to pirates boarding a US Naval vessel or even a civilian NOAA-operated ship. Let's put it this way, if anyone even gets too close to a USN ship, they get fired on. So if pirates boarded an American ship, they would likely end up at Gitmo or someplace similar.
By the reasoning you used above, Enterprise shouldn't have ever been able to defend itself, because that would be attacking citizens of a foreign power in unclaimed space, they shouldn't have been able to arrest and hold the pirates that attacked them in the Delphic Expanse, they shouldn't have been able to arrest and hold the Klingons who sabotaged their ship in the
Divergence story arc, or really anyone who wasn't a member of their own crew. That doesn't make much sense when I put it that way, does it?