Talk to me about CAT 6 in wall cable

As a network person CAT6 is too thick and hard bend or manage. For things like connecting a network DAC or network streamer I prefer fiber, this is of course assuming you don't need PoE for anything. I actually switched from CAT6 to fiber, and I think the fiber sounds cleaner. I really like simonb comment/idea.

Unfortunately, I think putting in fiber could be problematic. I have a long run over my house, which is two stories and the audio room is a dedicated room, which stands on its own at the farthest point from my router. I am not sure I can trust the installer to make the install.

Every ethernet cable sounds different. Get 2M runs of each one you're considering and have them terminated. Let them break in for 2 weeks anywhere in your network, and then do some listening comparisons. That will give you an idea of what you want to install. It may well be worth spending more at this stage.