How do I add certain Fields to a View (not all fields are showing in Add list)?
Is there some way to associate the View with a certain group of Fields


Yes, this is done by adding a relationship.

For example, if you create a content view and add a relationship of Content: Author, you will now be able to add user fields, such as User: Picture or User: E-mail to name a few.

Therefore, look if the already built in view is using a relationship. (It's under Advanced, located on the far right of the View's settings).