Hi . you can't run it to the cylinder as the Combi is mains pressure and would burst the cylinder .but you can definitely run a pipe to shower and basin .just tee off the pipe that runs to the kitchen tap. Then cap off the hot water cylinder and if the shower cold runs off the tank , cap off the tank and link up the cold main supplying the tank to the cold pipework outlet . then you will have equal pressure .
So if I did this it would mean that I could remove the cylinder altogether?
Yes. You could