Yes designed as a 24x7 device. If you want to you can use the Suspend mode (also trigger automatically via TV off) but not a must.
Yes, disconnect the Optical Cable and upload logs via MyOSMC → Log uploader so that we can check capabilities of your TV/AVR
Did you read this article?
Sounds like a HDMI cable problem. Suggest to try another cable or maybe we can see something in your logs.