Usually Shannon entropy is used to prove channel coding results. Even for source-channel separation results shannon entropy is used. Given the equivalence between Shannon (global) vs Kolmogorov (local) notions of information, has there been a study to utilize Kolmogorov complexity for these results (or atleast to replace the source coding part in source channel separation results)?
For channel capacity, it seems difficult to replace Shannon entropy by Kolmogorov complexity. The definition of channel capacity does not contain any mention of entropy. Using the Shannon entropy gives the right formula for channel capacity (this is Shannon's theorem). If you replaced the formula with Shannon entropy by a formula with Kolmogorov complexity, it would presumably be a different formula, and so it would be the wrong answer.
If you want to send a string with Kolmogorov complexity $K$ through a channel with capacity $C$ using only slightly more than $K/C$ channel uses, this is very easy. Find the description of the Turing machine that produces the string. Then encode it with an error-correcting code so this description can be sent through the noisy channel with only a small probability of error.
The hard part of the source-channel separation theorem is showing that you can't do better than the obvious method (described in the previous paragraph) of first compressing and then encoding. I don't know whether anybody has proved this for Kolmogorov complexity and channel capacity, but it's a reasonable question to investigate.
I am not sure what you are talking about when you use the local/global qualifiers on Shannon's entropy and Kolmogorov's complexity.
So correct me if I am wrong.
Shannon's entropy is computable. Kolmogorov's complexity is not. Therefore they do not describe the same problem.
You could see Shannon's entropy as an upper bound to Kolmogrov's complexity.