Value of power spectral density $N_0$ or effect of scaling bandwidth to SNR
The noise power continues to be $N_0/2$, independent of bandwidth. The reason is that the noise variance at the output of a filter with frequency response $H(f)$ is $$\sigma_n^2=\int_{-\infty}^\infty \frac{N_0}{2}|H(f)|^2\,df.$$ (This is a direct application of the Wiener-Khinchin theorem). Since you're assuming an orhonormal basis, the noise power is always $N_0/2$, independent of the bandwidth.

Then, doubling your signaling rate without a corresponding increase in spent energy results in halving the SNR.

Regarding your second question, $N_0/2$ is the PSD of the white Gaussian process assumed to exist at the input of your measuring devices. The PSD is indeed measured in W/Hz.

Note that it is a bit weird to define $P$ as watts per second. One watt is one  joule spent per second; watts per second would be joules, or pure energy. Usually, you assume you have a source providing $P$ watts; this source operates for as long as necessary (even from $t=-\infty$ to $t=\infty$).