Is it possible to hide a quick panel of the TeamViewer in the right corner and other notifications and pop-ups to enable a "silent" remote session?
Under the TeamViewer Options, Advanced, enable "Automatically minimize local TeamViewer Panel".
Other than that, there is no way that allows you to do so; it would otherwise be a security concern...
One possible way of doing this, without the TeamViewer Connection Panel showing up at all (with this the controlled computer cannot close the connection, unless they exit TeamViewer) is to use a 3rd party software or application such as this one.
For this specific program, you would need to change some of the settings on your TeamViewer to have it working properly.
In Windows, you will need to put the program in the Startup to work properly. Make sure to have in the TeamViewer Options, Automatically minimize local TeamViewer Panel disabled. There may be some bugs to it, but it is so far the best solution to hide the TeamViewer Connection Panel.
Make sure to read the Read_Me.txt file! It will give you a thorough detail about the program.
It is recommended to use this program on your own operating system(s).
With the program, you can connect to a remote desktop without the popup session panel like below.