By the Berry--Esseen inequality, the limit distribution of $Z_n^*$ is the standard normal distribution if $a_k=1/k^{1/2}$.
If $a_k=1/2^k$, then the limit distribution of $Z_n^*$ is uniform on the interval $[-\sqrt3,\sqrt3]$. This follows (i) because $(X_k+1)/2$ may be viewed as the $k$th binary digit of a random number uniformly distributed on the interval $[0,1]$ and (ii) $Var\,Z_n\to1/3$ as $n\to\infty$.
For the more general case with $a_k=t^k$ for $t\in(0,1)$, see e.g. this survey.
Using the formula (cf. e.g. formula 1.411.6 for $\tanh=(\ln\circ\cosh)'$) $$\ln\cosh x=\sum_{j=1}^\infty \frac{2^{2 j-1} \left(4^j-1\right) B_{2 j}}{j (2 j)!}\,x^{2j}$$ for all $x\in(-\pi/2,\pi/2)$, where the $B_{2 j}$'s are the Bernoulli numbers, we get all the cumulants and thence the moments of $Z_n$, in terms of Bell polynomials.