I haven’t heard about this – is this one of our plugins? Can you link me its page here on Meta? If it’s not ours (I don’t think it is), you’ll have to report this issue to the developers of that plugin.
I assume the “busy” status isn’t working as you’d hoped. Have you seen this: https://help.zoho.com/portal/en/community/topic/how-to-disable-chat-window-in-salesiq ? That seems to describe what you want to do
This sounds like it could be easily solved with custom CSS. I can’t access your forum, but try inspecting the page and finding the black overlay, so you can set it to display:none maybe?