What you probably want is hook_form_views_exposed_form_alter(). Inside this function, add your button using the form API, and then use #attach to add a JavaScript file. In this file, you can use jQuery to hide the button on page load, and then make it reappear when other values on the form change. Another method is to use Form API #states, which may be able to handle the JavaScript part of it for you. Best of luck!

Can you elaborate more on the "series of operations"? From what I gather, you have a View with exposed filters, the user fills out the filters, submits the form, and then you want your button to appear, before the results are shown?