Question about Mercykillers and arrests?

Yes, I would say that a Mercykiller, as a trained crimefighter, would be expected to put a stop to a crime in progress or pursue and arrest a criminal. Stereotypically, of course, there would be an amount of force employed that the other factions would find regrettable. Here's how I see the arrangement of checks and balances:

It's crucial to a lawful faction that the Mercykiller's act is distinguishable from performing random violence upon someone who ticks him off. Either he has a previous warrant or he doesn't. If he doesn't, then after the affray, there will be an investigation, possibly a cursory one but with at least brief detainment and a police report involved from the Harmonium laying out the facts of the incident and deciding not to press charges against the Mercykiller for one of the two reasons above -- or an arrest and short trial followed by the Fraternity of Order clearing the Mercykiller on those grounds.

The Hall of Speakers legislates, defining the laws on which the other factions can act. (They also need to be able to claim the Lady's support, and we have fluff pieces in the canon with the Lady specifically communicating with the factols requesting one or the other law enforcement activity.)

The Harmonium is a police force, with more power than the Fraternity or the Mercykillers to investigate, force entry, serve warrants, detain suspects and witnesses, and interrogate. Other citizens and other factions can arrest, but must turn prisoners over to the Harmonium on demand. On the other hand, the Harmonium cannot hold people indefinitely; citizens have the right to be charged or released. This step requires the Fraternity of Order's intervention.

The Fraternity of Order is the judicial branch. They do not pass laws; they interpret them. They can issue the warrants and judgments which legitimize the acts of the Harmonium and Mercykillers (and probably the Fated, in a less violent civil vein, and possibly the Doomguard in their role overseeing the Armory, though the Doomguard wouldn't care as much). They do not typically arrest, though as a special case a fair number of their officers have probably received Harmonium deputization. In the end, one of their people will be the "D.A." who decides whether or not to bring charges. Formal charge and a trial would be theoretically required before the Mercykillers can act.

The Mercykillers can hold people indefinitely, unlike the Harmonium, though only once they've been tried. They are the most "extraterritorial" of the factions (not counting Ortho or Mechanus), having a writ that extends beyond Sigil, primarily because they can pursue escapees and repeat offenders. Neither of the other two factions can impose their own imprisonment or executions; if the Fraternity or Harmonium kills someone, it's supposed to be in immediate defense of their own or others' lives, or an unavoidable consequence of the attempted arrest. Theoretically the Mercykillers can only execute a prisoner adjudged by the Fraternity, but by tradition they have considerable leeway in running the Prison and seeking bounties.

Finally, it is entirely possible that none of this is remotely as formal as all that, or that only the three factions care about the details of this arrangement, or that on the Guvners do. (However, bear in mind that if the Hall of Speakers wants to be effective in any way, they need an organization that will respect their edicts.)