Q1: I did not find the definition of a smooth stack in HAG2 (it would be nice if you provide a concrete citation!), but the definition of smoothness I know is: A morphism $X \to Y$ of algebraic stacks is smooth if there is a commutative diagram
$\require{AMScd}$ \begin{CD} U @>f>> V\\ @V g V V @VV h V\\ X @>>> Y \end{CD} where $U$ and $V$ are schemes (or algebraic spaces) and $f,h,g$ are smooth and $g$ is surjective. (see http://stacks.math.columbia.edu/tag/075U)
In particular, we can take in our example $X = \mathcal{M}_{FG}^{\leq n}$, $V = Y = Spec \mathbb{Z}_{(p)}$ and $U = Spec \mathbb{Z}_{(p)}[v_1,\dots, v_n, v_n^{-1}]$. So at least in this sense of smooth, the answer is yes: $\mathcal{M}_{FG}^{\leq n}$ is smooth.
Edit: As explained by Jacob Lurie in the commments, the map $g$ is not smooth and my argument fails. I am sorry for being careless.