How do I change the Date module to get 'days back and forward'? I want to restrict the dates a user can select when using past dates. I want a pop-up message when he tries to select a day +1 month from current date.

How can I do this? Do I need to override the module? If so, how?
You need to implement hook_form_alter() to add a new form validation handler. To avoid any conflict with the existing validation handler, you should probably remove the existing one added from the Date module, if there is one.