Let me try and answer a few of your questions. First, “success” for a customer success team is measured in the reduction of churn, which means that fewer customers defect from your service and more of them expand their use both in quantity (number of users, level of activity) and quality (more, especially sticky features used). You’ll have to build an understanding of what drives customer value and what diminishes it, and how to steer customer as well as your own company towards the former and away from the latter.
Second, there is a big variety in the scope of the customer success role. In some companies it is strictly defined as above, others have the customer success organization cover everything from onboarding customers to monthly or annual renewals or collections. In smaller companies the role would probably have broader scope, except where other teams cover a segment of the customer facing activity (e.g., technical support, sales for renewals and collections, professional services, etc).
Third, what works depends on your customer size and variety. If all your customers are on freemium or $10 MRR, then there’s not much personal attention you can provide, therefore you’ll have to build an automated system that will provide the customer success functionality. If your business is based on enterprise customers then you’ll have to provide a different mix of personal and automated attention, and if you have a broad mix of customers then you’ll need to define your cutoff points for the various segments.
Last, two companies seem to drive the discussion on customer success, one is Totango and the other is Gainsight. I strongly recommend you take a look at their websites and blogs for further information.