To hide the date navigation links on top of the calendar ("Today", "View Date:"), check the "Disable Event Month View Navigation Controls" option in the module settings.

To hide the view selection links under the calendar, check the "Only View?" option near the "Default Module View" selectbox. This setting does not affect the visibility of the "Add Event" link for module editors.

The look of the calendar is determined by so called themes, which are implemented as CSS files. You can select a theme per instance in the module settings. Try the pre-installed themes ("Default", "FixedBlue", ...), and if they are not what you are looking for, you can create your own themes: go to directory '/DesktopModules/Events/Themes/', copy one of the existing subfolders, rename it to e. g. "MyTheme", rename the containing CSS file to "MyTheme.css" and change it as you like. The next time you go to module settings, you can select "MyTheme" from the themes selectbox.