Connect cable modem/router to different circuit than audio?

The router should not be on the same circuit as your audio.
A router with the coax line feeding it from outside generates high levels of RFI which translates into noise, an audible amount of noise.

Since you can’t isolate your audio from the electrical devices you mentioned, a power conditioner or power strip with filtering would be in order. Brick Wall or Surge X are very effective conditioners or a Furman Power Station will provide filtering from RFI/EMI noise on your shared AC line.

You have a very nice system, isolating it from the noise-generating electricals will provide noticeable benefits.

You're welcome. Let us know if you hear any improvement. 

You should move the router to a separate line. Although, it would be good if you could verify that this line isn't shared by your audio. You can shut off the breaker to confirm this.