you are still not understanding. Again, what you think should happen, is not designed to happen that way.
A - that chat which you are referring to as Group Chat - is in fact a One on One Video chat between you and the person where you clicked the film icon (the film icon signifies video communication). It looks exactly like the Group chat - yes. But it is a one on one conversation. When you type something in the message field - the user will get a popup that you want to chat with them. Because it is a video chat, it does not pop up in the member menu like you want it to - that is just the way it is designed and that is the only answer i can give you.
B - What you are looking for is the simple messenger - make sure this module is installed in your admin section
C - the simple messenger is only accessible from the users profile page. Once installed, if the user is online, then an open field appears for you to send them a quick message - at this point, the message window you are looking for will appear.
Again, it is the design of it, and while we all may have our only opinions, for whatever reason, thats what they decided to do. So, until they change it, or you come up with a workaround, you are kinda stuck with the functionality