Let $\sigma:=\mathrm{Re}(s)$, and consider the principal branch of the logarithm. For $\sigma>3/2$ we have $$\begin{align*}p\log(1-p^{-s})-\log(1-p^{1-s})&=p\left(-p^{-s}+O(p^{-2\sigma})\right)-\left(-p^{1-s}+O(p^{2-2\sigma})\right)\\&=O(p^{2-2\sigma}),\end{align*}$$ hence the "Euler sum" $$H(s):=\sum_p\left\{p\log(1-p^{-s})-\log(1-p^{1-s})\right\},\qquad\sigma>3/2,$$ converges absolutely and locally uniformly. This implies that the Euler product $$ G(s):=\exp(H(s))=\prod_p\frac{(1-p^{-s})^p}{1-p^{1-s}},\qquad\sigma>3/2,$$ defines a non-vanishing holomorphic function. In the original half-plane $\sigma>2$, we have $$ G(s)=\frac{\zeta(s-1)}{F(s)},\qquad\sigma>2,$$ hence $F(s)=\zeta(s-1)/G(s)$ extends to a meromorphic function in $\sigma>3/2$ with a simple pole at $s=2$ and no other pole there.
Regarding your second question, I am not aware of any papers where this function was studied. My argument above is rather standard though.