If I have an object like Account , Contact , or a custom object , is it possible to add a custom button (or custom Action) on the Edit page ?
Custom Buttons and Object-Specific Actions do not appear on Object Record Pages when in Edit mode. They are intended to be used from Detail or other Pages to automate the creation of new records, opening detail pages in edit mode with pre-defined values populated, opening records for editing in some context, auto-updating records, or performing other types of actions such as opening an external web link.
An Edit page is already in "Edit mode", so clicking a button or action from within one could easily create conflicts with the controller for the object. Standard buttons for the object's controller (or a custom page controller that's been substituted) need to be respected and observed in those situations. That isn't to say that a custom button defined for use on a custom page to work within the confines of the custom controller couldn't be utilized while editing the page; provided it called a method contained in the page controller.