Are there any known examples of $ ((n,K,d)) $ codes with $ d \geq 2 $ for which it is not possible to find a basis of codewords that are stabilizer states?
A code word stabilized (CWS) code is defined in https://arxiv.org/abs/0708.1021 and is of this type where, in particular, the different stabilizer state codewords are related by Paulis.
Even though the paper is from 2007 this seems to be the state of the art for non stabilizer codes. At least I can't find any newer papers on the subject.
The paper claims that all known examples of "codes with good parameters" are CWS codes. What is an example of any code (I don't care what the parameters are as long as $ d \geq 2 $) with codewords that are not stabilizer states?