I have done all of that, but now almost want to send the darned thing back! In order to get the sound through the amp, I had to change the Panny DP-UB820 over to HDMI 1 (video) and 'Blu-ray Disc' input on the amp (audio). THEN I got TV Audio through the amp. Amazing and quite outstanding Dolby Vision and Atmos! HOWEVER, when wanting to watch a BD, Noticed that the picture would only work through HDMI 2 and the sound through HDMI 1. (Frustrating!) I know I could just use the one HDMI input for both sound and vision, however, Panny says it can do it, so it should! And I wouldn't be surprised if that still didn't work! I suppose it's just that HDMI 2 is the Arc/eArc input? Sky TV works just fine through either. Although the 'fairly new' Sky Q box does need to be reset two or three times a day! Every time I turn it off and then back on again later, it refuses to put a picture up until you turn it off again, when you get one quick flash and that's that! So I have to turn the box off at the wall and then back on again, which is getting rather tedious. Sky tells me that's what I have to do and seem unable to fix the problem, without sending out an engineer! My old system 'mostly' worked for years. I'll try the single input tomorrow (Saturday) in hope that it will work, though I'm sure that I will have to give up my wish for 'proper' sound on the TV apps. That is, unless any of you guys know better?
So, I have some similar gear to yourself.
I have a Panasonic UBP9000, Sky Q, games consoles and an A4A receiver. The receiver hasn't had an update to enable VRR (yet) and so no games console runs via that. I do have the second HDMI audio out on the UHD player connected to the receiver though.
The only issue I have is from time time the audio from the UHD player will come out via eARC, there's obviously conflict there among the devices over which one takes priority. So, I have to manually opt for the correct input on the receiver for sound from the UHD player.
Otherwise, its game console into HDMI 1, receiver into HDMI 2, UHD player into HDMI 3 and Sky Q into HDMI 4. HDMI sound is set to passthrough on the TV.
I don't have any of the issues you describe I'm afraid. All I can suggest, and I assume you've done it already, is turn everything off at the wall, connect it how you want it, presumably similarly to mine, turn it all on and maybe consider resetting everything? It sounds like it's nothing more than a handshake issue somewhere, nothing terminal.
Personally, I'm not afraid of resetting anything and when I have a problem, my first option is always to power cycle everything as HDMI is a finnicky and demanding mistress.
Good luck!