My 3 Android/Google TV devices work very well with the integration. I have 2 Sony Bravia TV’s and a random brand named RCA.
With WOL disabled when you put the TV into standby mode it also shuts the network port. The only solution is to enable WOL. It is indeed odd behaviour if your TV is switching on randomly, very likely it’s a bug in the firmware you have installed. Worth looking into the release notes, if you can find details of your firmware. I hope you’ve already done so, but have you updated to the latest available firmware?
An alternative could be to use a Smart plug to control switching the TV on and off. But that would only really work if it doesn’t go into standby as a default.
In terms of the current channel being available in Dashboard cards, I don’t have any of my TV’s directly connected to an external source. I have a separate box for satellite TV. I can see the source being used though, such as Sky Q or HDMI 4 and can control the volume. Also apps are displayed, such as Plex or YouTube.
I don’t know if it’s the latest firmware, it doesn’t find anything from the TV and I don’t understand if there’s a site online that contains all the firmwares.