Joshua,
1. "Customize this page" will appear if you are loged in as site admin.
2. Changing the layout of the individual group/event page:
Can you give an example of what you would like to change?
Those pages in most of the cases uses base oxwall styles ( that are applied sidewide). So to completely change the layout requires a lot of custom work. You will need to create new .html template for needed plugins ( ex. events), and add new styles.
But if you would like to change the look and feel of only several components within the plugin, it is possible via CSS. All depends on what you would like to achieve.