$y = y_0 e^{kt}$ solves $\dot{y} = ky$ because the exponential is its own derivative up to a constant.

True

True. $\frac{d}{dt}(y_0e^{kt}) = k y_0 e^{kt} = ky$, which is exactly what the equation demands. Any quantity whose rate of change is proportional to its size — compound interest, unchecked population, radioactive decay — follows this form, with the sign of $k$ deciding growth or decay.