HTPC to Pioneer AMP - no sound

Off hand, I would say "you can't get there from here". When you use an HDMI port that is expecting both Video and Audio. With what you have, it appears to only be Video in the HDMI. Also when you connect HDMI, in most cases (but not all) the HDMI becomes the "default audio playback device" and thus there is no audio out from the regular PC sound devices.

Check the Control Panel/Hardware and Sound and finally the Sound panel and see what is listed in the Playback section. If you only have the "Speakers" that is your sound card. But if HDMI is also listed, change it to the Speakers and then see if you get audio out from the PC's sound ports.

As far as whatever you are connecting the HDMI to, usually if you select the HDMI input then that becomes the source for both video and audio and there is no option to have HDMI video and audio from some other source at the same time.

As I originally stated, "it looks like you can't get there from here" with the hardware you presently have. It would appear you need a separate video card that will provide both HDMI video and audio (the video card has it's own "sound card" on the video card only for HDMI audio). With a slim case and a 135 or 180 watt power supply (what is listed in the specs) this can be a problem. The slim case models use a special size power supply and not a standard ATX style and what is available is limited. The slim case models use "Low Profile" type add on cards.

Best advice, if you can return the PC, you can then get either a full tower that you can add the needed video card or buy one that has that included.

Finally, we list our system specs in the "My System Specs" on this forum, not a link as you provided which is not what you actually have, but what can be ordered with the PC.