I don't know much about the effects of anesthetics, but I have taught basic nerve function at university level, so I hope I can help a little with that part.
Potassium flow does not start an action potential - on the contrary, increased potassium flow makes it harder to depolarize the membrane and initiate an action potential.
A brief explanation on membrane potentials and action potentials:
During rest the flow of potassium is much higher than the flow of sodium (and also chloride) and hence the membrane potential is driven towards the Nernst potential for potassium. The resting potential is often somewhere between - 40 to - 80 mV depending on cell type. To initiate an action potential the membrane potential needs to depolarize enough to exceed the threshold for opening voltage-gated sodium channels. The ion flow across the membrane will then be dominated by sodium and the membrane potential then moves towards the Nernst potential for sodium (which is positive, maybe 40-50 mV). These voltage-gated sodium channels will then close shortly after and then stay closed for some refractory period where they can't open, which brings the membrane potential back to the resting potential, because the flow will then again become dominated by potassium flow. There are also voltage-gated potassium channels in the membrane that open with a slight delay compared to the voltage-gated sodium channels. The opening of these potassium channels result in a faster return to the resting potential after the peak of each action potential.
Regarding your question on a small (but insignificant) decrease in potassium conductance:
Decreased potassium flow will bring the membrane potential further away from the Nernst potential for potassium and closer to the Nernst potential for sodium, i.e. causes a slight depolarization, which makes it slightly easier for the membrane potential to exceed the threshold for opening voltage-gated sodium channels and initiate an action potential. If the flow of potassium is completely blocked, the membrane potential would be determined by the relative permeability of chloride and sodium.
The Goldman equation is the key to understanding membrane potentials, and relative permeability changes between the potassium, chloride and sodium ions (and in some cases calcium ions) is what drives an action potential.
Neuroscience by Purves et al is a good resource on all of this.