Hi Gunter,
Welcome to the world of browser differences.
As soon as you start customizing client side resources, you must also prepare yourself for having to dig, debug and test the different browsers for different reaction to your changes, and this case is no different.
There are various tools that you should learn how to use, like FireBug for instance, and for IE, you can use script debugging directly inside Visual Studio. There are also quite useful websites with references and difference reports between browser and the one I most often use is www.dottoro.com, or more precisely, their reference at help.dottoro.com. This is a great reference site. Also, W3Schools is a good reference site.
In this particular case, I simply started the application inside of FireFox and set breakpoints inside the scripts on some key functions and soon found out that FireFox does not fire the oncontextmenu event, rather it simply fires the onclick event for right clicks on the RichTextBox. In order to fix that, cross browser (hopefully), I think it would be the simplest fix to cancel the event bubbling of the click event within the RichTextBox_Click JavaScript method. We are already handling the click event, which means we don't want it to bubble up the DOM and fire on some other element anyway. In order to do that, add the following as the last statement within RichTextBox_Click:
mobjApp.Web_EventCancelBubble(objEvent);
Hope this helps,
Palli