Odd-bit primes ratio

Yes. This was proven in

C. Mauduit and J. Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Ann. Math.

I found this by searching for "evil prime" and "odious prime" in the OEIS. More precisely, they prove the Gelfond conjecture:

Let $s_q(p)$ denote the sum of the digits of $p$ in base $q$. For $m, q$ with $\gcd(m, q-1) = 1$ there exists $\sigma_{q,m} > 0$ such that for every $a \in \mathbb{Z}$ we have

$$| \{ p \le x : s_q(p) \equiv a \bmod m \} | = \frac{1}{m} \pi(x) + O_{q,m}(x^{1 - \sigma_{q,m}})$$

where $p$ is prime and $\pi(x)$ the usual prime counting function.

Yes, the ratio approaches 1/2. This was proven in

C. Mauduit et J. Rivat, Sur un probléme de Gelfond: la somme des chiffres des nombres premiers.

See Three topics in additive prime number theory for exposition. Also, the poorly-named sequences in Sloane: A027697 and A027699.