Solved – ACF of a stationary time series

1. It depends on what you mean by "drop to zero relatively quickly". For a stationary AR(1), $y_t = rho y_{t-1} + epsilon_t$, $|rho| < 1$, the ACF never goes to zero, but rather decays as $rho^k$. In contrast, for an MA(1), $y_t = alpha epsilon_{t-1} + epsilon_t$, the $E[y_t y_{t+k}] = 0$ for $k > 1$, so it drop to zero very quickly.

2. Ljung-Box is not used for testing stationarity. Rather, it is used to test for the presence of serial correlation. For example, an AR(1) is stationary, but the Ljung-Box test will reject the null hypothesis due to the presence of serial correlation in the AR(1). If you apply the test to a non stationary time series you will also reject the null, but you can't conclude that the series is non-stationary just because the Ljung-Box test was significant.

Just a side note, an iid time series is still stationary (trivially), so just saying "if the series is stationary I'll reject the null hypothesis" is not correct.