Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms

For $(1)$, I believe the size would still be $\gg x^{1-\epsilon}$. However, if you allow $$\sum_{p\sim x} |A(p,1)|^2+A|(p^2,1)|^2$$ a lower bound of similar sort is obtained by Blomer-Maga's paper Corollary $4.3$. (In any case, one elementary way to know the average size of the Hecke eigenvalues is by writing them in Satake parameters using Shntani's formula.)

For $(2)$, the Dirichlet series appears in the Rankin-Selberg convolution of $$\langle \mathbb{P}f,\overline{\mathbb{P}f}\rangle,$$ where $\mathbb{P}$ is the standard projection operator from $\mathrm{GL}_3$ to $\mathrm{GL}_1$ which comes in the definition of global zeta integral. For instance, see chapter $10$ of Goldfeld's automorphic form book. Thus one can obtain functional equation and meromorphic continuation from the above.