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Are you connecting over lan or wan?

On the remote host computer, does the blue tray icon show a blue dot, or gray?

On my system, it seems the hostname is picked up from the direct connections field under the properties of each individual connection. To test, right click on any connection, choose properties, click the radio button for direct connections, then enter the details of direct connection. If using internet id, click that radio button again before closing properties. Hostname should now show what you entered for direct connections.

I can offer a couple of suggestions. No guarantees.
As for the options window in Firefox, right click the connection in viewer, select properties, and under remote screen-additional properties, make sure "Capture Alpha-Blending Windows" is checked.
As for the mouse lag, I have this happening on an XP box that I maintain. The only solution that I have found, is to run Firefox with "Low" priority. To test this out, open task manager, find the Firefox process, right click on the process, select set priority, then select low. You can do this by starting Firefox with this shortcut, if the test has the desired result. This path is for an XP machine, you may have to edit it for Win 7.  
%windir%\system32\cmd.exe /c cd "C:\Program Files\Mozilla Firefox" && start/low firefox

Hope this helps !!!

Try this and see if it helps.
Using viewer on client machine, select host machine and right click. Select the properties at bottom of list. In properties window, select "Additional Properties" on left side. Make sure "Capture alpha-blending windows" is checked. Hope this helps...