I'm not familiar with your particular models but here some educated guesses.
No sound wuth these connections
Blu-ray > HDMI 2 on TV
Blu-ray Audio > HDMI 2 on 5600
My guess is this is a setting in the player. You probably have to tell your player to use the HDMI audio only connection. My guess is the sound is going to the TV. In theory these should work if set to some auto type setting ut in practice not always.
Blu-ray > HDMI1 on TV
Blu-ray Audio > HDMI 2
No video makes sense because the blu-ray video is not on HDMI 1 for the TV and the auto feature of theplyer isdetecting the 5600 on the audio only connection.
I think you have a few problems here and some possible solutions.
1. I'm not a 100% on this as you will have to check for your specific model but many of the Vizio TVs only get full 4K with HDR on HDMI input1. The other connections can only do basic 8bit 4K.
2. If HDMI1 is your ARC connection you can't have full 4K/HDR and ARC at the same time unless the sound bar or receiver is full 4K capable and all the 4K/HDR devices are switched through the sound bar or receiver. The 5600 is not full 4K/HDR capable.
With an Apple TV and your blu-ray in the system do you use the apps in your TV?
I would get an HD Fury Integral 2 or HD Fury Vertex for your Apple TV and Blu-ray player.
If you want all your devices (Xbox and Dish as well) to have full 4K capability then you need aa 4K switch which has an HDMI audio out capability. One comes to mind but HDFury has a new product shipping in January that would work.
The HDFury Maestro. You only need the Central unit but I think you get both now for $499. After the Pre-order both cost $999 and the central unit will cost $599.
This should also work but haven't used it.