Phillips curve and flexible prices I am working in a New Keynesian context so that the Phillips curve is usually specified as follows 
\begin{gather}
\pi_t=\beta E_t\pi_{t+1}+\kappa x_t
\end{gather}
where $\beta$ is the discount factor, $\kappa$ is the slope of the curve depending negatively on the degree of price stickiness in the economy and $x_t=y_t-y^N_t$, with $y^N_t$ is the natural (potential) level of output

What is driving me crazy is if this curve, specified as above, still make sense when we assume prices are fully flexible. 

I would say no, because in a flexible price context, the output gap would be zero and the firms would optimize in a static fashion (no need of taking into account expectations of future inflation). But, I have no idea about how to "transform" the model in this direction.
The New Keynesian Philips Curve (NKPC) may be derived using price settings frictions introduced by Guillermo Calvo. The NKPC can under such conditions be written as you write it, i.e. $$\pi_t=\beta E_t\pi_{t+1}+\kappa x_t,$$ and where $\kappa=0$ if no firm change their prices, and $\kappa=\infty$ if all firms change their prices.  (Technical note: Here I allow myself to work with the extended real line, viewing $\pm\infty$ as entities and not concepts.)

The background given above, then, does not seem to say anything about the output gap $x_t$. If $\kappa=0$, then inflation follow its trend, i.e. $$\pi_t=\beta E_t\pi_{t+1},$$ and if $\kappa=\infty$ then $\pi_t=\infty$ (given that the expected inflation is bounded below).

This, at least, answer your first question: If fully flexible price setting is interpreted as the case when all firms change their prices, then $\pi_t=\infty$, which is understandable, but not realistic. What we may say, though, is the more firms there are who change their prices, the closer to hyperinflation we are.

To me it seems that we can not from the NKPC alone say whether the output gap will be zero if $\kappa=\infty$. Often we have to supplement the NKPC with the DIS $$x_t=E_tx_{t+1}-\frac{1}{\sigma}(r_t-r^n_t),$$ where $r_t$ is the real interest rate and $r^n_t$ is its natural level, and add some additional information on whether we have random shocks in e.g. the output gap.