Is it possible for a coded message to contain characters that will pass every test for randomness?

You could add codified thermal noise over it when you send it and the receiver subtracts the noise when he gets the message. both sender and receiver need to have an identical copy of the noise.

yes it can.

you could also use an Ottendorf Cipher

The Ottendorf Cipher sounds similar to @ragingloli’s idea of adding thermal noise to the message. Either way it would seem that additional information is required to make sense of the randomness. I am almost but not entirely convinced that either method would produce a purely random sequence.