is there weather in the tropopause?
Yes, but not often. See the following Wikipedia paragraph about that.
The tropopause is not a "hard" boundary. Vigorous thunderstorms, for example, particularly those of tropical origin, will overshoot into the lower stratosphere and undergo a brief (hour-order or less) low-frequency vertical oscillation.[9] Such oscillation sets up a low-frequency atmospheric gravity wave capable of affecting both atmospheric and oceanic currents in the region.[citation needed]
Most commercial aircraft are flown in the lower stratosphere, just above the tropopause, where clouds are usually absent, as also are significant weather perturbations.[5]:21