That ebay listing ran out, so I would need a model number or photo to see what you see on your end. I'd say the switch is part of the trouble though, since direct connects of equipment around the switch seem to work consistently.

There's no way unfortunately to upgrade your old JVC home cinema unit. FWIW, Today I saw a Yamaha HT receiver with all the trappings for < $200 US. Might be time for you to explore that option, and make things easier for your setup/entertainment needs.

Playback of MKV files bring in a whole other set of issues, since they are all not the same (MKV is a container). I'm not even going to go into writing an essay on this topic it's so complex. Combined with the fact that you are working with an ancient home cinema system, I would probably rule out any fix in this regard. Some simply will not play, others may have some amount of audio video sync issues. FWIW, I added a WDTV Live media streamer box to my older setup, and it seems to handle most file types/codecs quite well. A simple laptop or PC would handle these files even more thoroughly.

Have you used the following steps from the XBOX site to adjust your audio output settings(?)

There may be something similar that is required on your HD Sat box. Look in the user manual to see if stereo vs DD5.1 require anything specific when you utilize one type of cable vs another. YMMV. I'm not certain whether or not you can point a finger at the JVC in this case either since it's designed to accept and interpret what's being sent to it via the digital optical port in back.