Is it possible to insert programatically my own error in the dialog, so that it is shown and prevents he user to start the edit session? (Description)
You can't add you own message to this dialog but you can create an event listener to watch for the start editing event to fire. When this event occurs you can then run some code and display your own message dialog.
There are several ways to implement the event listener. One way would be to create a custom extension that tracks events when it is turned on.
I would develop an ArcMap extension with your own toolbar. Generate two buttons for start editing and stop editing. Additionaly you can integrate any ESRI standard tool very easyly. After clicking “start my edit session” you can add your field, make all necessary checks and then start the edit session. The end user of your product now should use only your toolbar (for starting and stopping edit sessions) and should forget the standard ESRI toolbar. This approach would additionaly help you with your other post here concerning the event handling of edit session.