Hi!
I just installed the program, I’m getting to know it, studying the settings.
Can the program change the state of the icon in the system tray/taskbar when there are/are no unread messages?
By changing the color or animation? Without using standard Windows notifications.
I found only a sound notification and a notification in a pop-up window.
Add
Answering myself 
For the system tray there is only a notification about a new letter.
There is no permanent notification about unread letters in the system tray.
There is no permanent notification about unread letters in the system tray. Only new mail.
Once a new unread mail message in the Inbox “has been clicked in the Inbox”, then the system automatically removes the new mail envelope icon notification in the system tray. There is no way to permanently keep it there.
If you then want to see all other unread new email, then “when the program is open” eM Client has an Unread folder under Favourites top left to show those. You can enable to show Favourites via "Menu / Settings (Preferences) / General.