Methane is a much more powerful greenhouse gas than CO2, so that's going to be as much, if not more, of a contributing factor to the world's warmth.
As long as you have water, anoxic or not (or liquid ammonia, or liquid methane, or whatever), then yes, there can be Earthlike weather. Extreme weather events will generally be less violent than on Earth if the world receives less solar power--so, e.g., you may have hurricanes, but they won't be as large--but rain, fog, mist, and lightning don't take much, and the world would have to be pretty darn cold before you would eliminate the possibility of thunderstorms and tornadoes. Even Mars has tornadoes (pretty big ones, too, since the air is less dense, so it takes less energy to make them).
It's the 5% nitrous oxide, though, that seems like a deal-breaker to me. Trace amounts of oxygen? That'll react with methane fairly quickly, but if it's being continually replenished by the native life somehow, sure. Incidentally, nitrous oxide is also a far more powerful greenhouse gas than CO2, but it's also a powerful oxidizer, and will react with methane to form nitrogen gas, more carbon dioxide, and water. So what's producing so freaking much of it, and why? And what's simultaneously replenishing the methane? 'Cause if something is constantly pumping out N2O, but not any new methane, eventually that methane fraction is going to drop, precipitously. And what is preventing the native life forms from taking advantage of such an obvious metabolic free lunch? You'd think some microbe would evolve in a geologic instant to generate energy by just breathing in copious free methane and nitrous oxide and catalyzing their reaction, eliminating one gas or the other from the atmosphere even quicker.