Mazda 6 V6 auto rough misfiring – fuel flow problem?

2007 Mazda 6 3.0L V6 automatic with ~80k miles:

1. Runs "rough," especially while stopped in drive – feels like what I imagine an occasional cylinder misfire would feel like, producing a sort of an unbalanced vibrational shudder.
2. Recently, in addition to the above, full-load full-throttle feels like it is sporadically cutting fuel to the engine even nowhere near redline and with no traction-control indicator. (Traction control is functional and its light illuminates when it is intervening.)
3. Behavior has persisted over several tanks of gas. Occurs when engine is both cold and warmed up.
4. Check engine light is functional but has not illuminated.

My naïve guess would be there’s a fuel flow problem. I don’t know how to diagnose. If the check engine light is not on could there still be engine codes that would be helpful? Is there a fuel filter that should be replaced? Is there a fuel pump and a way to check that?

---

Update: I have confirmed that the only fuel filters are in the fuel pump itself. I also found suggestions that driving tanks empty, which I often do, can reduce the pump life by giving it less cooling. Fortunately there is only one fuel pump and it can be replaced without removing the fuel tank. Since it’s a $70 part I’m going to try replacing that and note what I find here….