You mean the Host icon in the system tray, correct?
If so, I'm afraid this cannot be possible. The icon is part of the program branding, besides it must be immediately recognizable due to security reasons and be associated with the program that runs behind it. No doubt antivirus software companies will frown upon this ability, should we start to provide it.
Also, will the beta update be available via the in-app update check, or does it have to be downloaded manually?
Usually, we do not enable automatic notification about beta updates, only stable ones. But the beta version will be publicly available on the site so anyone can download and use it. We will be looking forward to receiving feedback from beta testers.
Your settings are correct. This is by program design - when you move your admin mouse pointer the remote mouse pointer also moves, but not vice versa. So inevitably if you once move your remote mouse pointer (by physically moving the mouse) and then the admin moves hers the remote pointer will "jump" to the admin pointer location.