I use bronze bore brushes and zinc-coated brass jags on the barrels of my P-07's. After a range trip, I soak them in a jar of Hoppes #9 bore solvent for many hours, usually overnight, to help remove carbon and copper fouling. My barrels are in pretty decent shape, each having a few thousand rounds through them. I have an inexpensive bore scope I ordered off of Amazon to check my cleaning work and the condition of the bores.
Personally I think the nylon bore brush that comes with the P-07, and nylon bore brushes in general, are bordering on worthless. I have some and will occasionally use them when the barrel has Hoppes solvent or some other copper remover that will react to bronze brushes. They really don't do much, compared to a bronze brush. On the other hand, the slotted bore cleaning rod that comes with the P-07 works just fine with patch cloths. It's not a jag but it works. I also use mine to retrieve my barrels out of a jar of Hoppes solvent.
I have removed my extractors and cleaned behind them on both pistols, but I wouldn't bother with that until you have at least 1-2K rounds though it. One thing to keep in mind is that you don't want any wet lube in there as it will goop it up more quickly. Spraying it out with brake cleaner or similar is probably not a bad idea, but it isn't going to get all of the goop out. When I have removed the extractors in the past, to clean them and the extractor channel, I have spent quite a bit of time and effort removing the carbon / powder deposits out of there.
I never had any out-of-battery issues with my P-07's. I can induce light strikes with Russian ammo using my CGW 15 lb hammer spring, and I can cause a failure-to-eject by dropping the mag and firing the round in the chamber, but otherwise my guns run flawlessly no matter what. Well I also found a compatibility issue with one obscure type of ammo. Really though these guns are solid performers overall.