The answer to your first question is no: $D(Q||P)$ may be however large while $D(P||Q)$ is however small. E.g., let $P$ have masses $s$ and $1-s$ at points $0$ and $1$, respectively, and let $Q$ have masses $t$ and $1-t$ at points $0$ and $1$, respectively, where $0<s,t<1$. Then
\begin{equation}
D(P||Q)=s\ln\frac st+(1-s)\ln\frac{1-s}{1-t},
\end{equation}
\begin{equation}
D(Q||P)=t\ln\frac ts+(1-t)\ln\frac{1-t}{1-s}.
\end{equation}
Let now $t\downarrow0$ and $s=te^{-1/t^2}$. Then $D(P||Q)\to0$ whereas $D(Q||P)\to\infty$.

The answer to your second question is yes: Pinsker's inequality states that
\begin{equation}
\sup_A|P(A)-Q(A)|\le\sqrt{\tfrac12\,D(P||Q)}.
\end{equation}