Hi Ajay,
> 1) Is there a way for it to save the project on shutdown without showing the message box dialog, but instead just saving the project file automatically?
Use the /SilentMode /exit command line options ("TestComplete Command Line",
http://www.automatedqa.com/support/viewarticle.aspx?aid=10997)
> 2) Silent mode still shows the mht file at the end of the TestExecute run. Is this correct? Can it be stopped? Again, I am manually closing this screen.
Right-click TE tray icon, select Options and disable the 'Activate after test run' option under Engines|Log node.