Channel coding results using Kolmogorov complexity
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.