Consider using an HDMI switcher. Wait, first try running the cable box with an HDMI DIRECTLY to projector (no receiver in between). If this works, then get a switcher. Or you can also try component output on cable box to receiver, and have receiver transcode to HDMI output, and maybe that can work.
This is likely an HDCP handshake issue, and while it can be the fault of any of the in line components, it is most likely the STB itself.
Try the above two experiments, and report back. For your audio, just run SPDIF from cable box to receiver, if any of the two experiments work. Yes, it's not quite as sexy, but even with the switcher, all you need to do is hit one extra remote button, and that's assuming you don't have macro programming to begin with.
Hope this helps.