Is $E′(k,m)=E(0^n,m)$ semantically secure?
The algorithm $E'(m)=E'(k,m)=E(0^n,m)$ is defined with a hard-coded key, thus the key is part of the algorithm definition of $E'$.

Because of Kerckhoff's principle we generally assume the attacker to know our algorithm definition.

Because of this, the attacker can just try decrypting the challenge ciphertexts of the eavesdropper security game himself (or try encrypting the challenge messages himself if encryption is deterministic).

The choice of the messages $0^n$ and $1^n$ is entirely arbitrary and any message would suit the purpose here with these two having the advantage of the shortest possible description (instead of something like $0^{n-1}||1$).