Have I found the correct Emission Price
To calculate the socially optimal level of $e$, you should look at $\pi_W+\pi_H$ (presumably that's also how you come up with the socially optimal $x$). At $x=20$, $e=1$. 

The intuition is that the woodworker's production creates an externality of $-2a=-x$ on the hotel. Since the hotel has to right to ban noise pollution, the woodworker would have to compensate the hotel for its cost from putting up with the noise. At social optimum, the marginal compensation ($e$) should exactly cover the marginal external cost ($-1$). Hence $e=1$ at the social optimum.