If a field is not required at the application object level, can I make it required at the form level?

Yes. If an application object field is not a required field, you can still make it required in your form. First, build a form using the existing application object. Double-click the field that you want to make as required. Select the Required Field check box in the Field Properties dialog box. Save and check in the form. When the form is in use, the user is required to fill data in this field.