I don't see how CB could possibly be causing that unless your 3rd party extension is a CB plugin and isn't using latest CB API. CB doesn't interact with anything outside of itself and nearly exclusively uses our own APIs instead of Joomla's. It also can't style anything outside of itself due to our CSS being namespaced.
In the back end, when creating a View for a list, there is a permissions tab, and under that tab one may create a permissions profile for both the front and back ends of the website
I'm not sure I understand what you mean by this. Can you be more specific? You mean CB > List Management > Permissions? Those permissions are for backend management of userlists they've no impact on frontend.
When a view into a list has been created and one views the list in the front end, a user with permissions should see both a “New” button, and a “Delete” button (assuming permissions for those to processes has been enabled).
CB userlists don't have a new or delete buttons. I assume you mean in ContentBuilder? Sorry, completely lost here. Please attach screenshots with further explanations.