Probability of detecting errors in codewords
I think your approach is fine, but the question is not fully well-defined (unless I'm missing something). 

The general error probability is 
$$\sum_{i} \Pr(C_i\text{ is transmitted})\Pr(\text{error}\mid C_i\text{ is transmitted}).$$
The questions explicitly says that the error probability of all codewords except $C1,C2,C3,C4$ is negligible, so the error becomes:

$$\sum_{i\in\{1,2,3,4\}} \Pr(C_i\text{ is transmitted})\Pr(\text{error}\mid C_i\text{ is transmitted}).$$

Computing the error probability is easy: if the Hamming distance is $x$, it takes $\lfloor x/2\rfloor +1$ bit-flips (or more) to corrupt the codeword, and $x$ bit-flips (or more?) to be undetected by the receiver. You can easily compute that in a similar way to what you wrote, and get a reasonable bound.
However, the question explicitly says that the terms $\Pr(C_i\text{ is transmitted})$ are unknown, yet it requires you to give an exact numerical answer. contradiction.