What makes The Fed (or even US) decide to intervene militarily in some affairs, yet ignore or do little in others?
Don't have a great answer to that question. In the end, the Prime Directive seems like it's not any different from other rules in that it's open to interpretation. Every officer must decide how and when it actually applies, knowing that it's often easier to ask forgiveness than permission. Sisko's actions could have been interpreted as a violation of the PD: OTOH, the Klingons were stationed at DS9 just before they went into Cardassian space, so Sisko could have argued that they'd already dragged (albeit indirectly) Starfleet into the situation.
The discussion in the ward room is interesting. Dax doesn't say that they're violating the PD by warning Cardassia. She says, "The Klingons are still our allies. If we warned the Cardassians, we'd be betraying them."
O'Brien follows that with a comment about the Klingons being right. Whether that's because he actually believed it or because of his own hatred of Cardassians is something I don't know the answer to. Regardless, it's clear that no one sees helping Dukat as a violation of Federation rules. Like anything else in fiction, the PD is probably just another plot-device used to advance the story rather than being an actual rule.