As the question said, I have added Widgets like Catalog Products List and cms blocks using admin WYSIWYG editor. While adding them, I am always selecting the custom theme I have created. Are these changes getting saved in the theme? If I were to copy this theme and paste it into a fresh Magento installation, will all the changes be reflected? If not, how to go about it so it does?
Your admin changes saved in database so if you move only theme into new fresh magento then nothing happen.
So, You have to migrate all database changes along with theme else you have to do all changes again manually.