How can I minimize a chrome browser session to the tray so that it remains there permanently even a?

There is no direct way to do this, because all processes that take place on your pc is stored in RAM, which loses all its content when power is lost (computer off), which includes Chrome. However, there is a workaround:

1.) Go to Chrome Settings
2.) In the second group named "On Startup"
3.) Choose the option that says "Continue where you left off"

This way, when you open up Chrome, you will have all the tabs you left off with. (hence the name of the option)

You can also drag the shortcut to Chrome onto a blank space in your taskbar, and it will stay there, but will not be running when you start up your computer.