1. Allow only members of group 'x' to register for the events?
You can not restrict registration for any group of members outside of the distinction of public / private. Private just means you must be a logged in member to access registration.
3. Apply a discount for an event for members of group 'y'?
You can apply discounts based on user selections in your custom fields, or you can also use discount/coupon codes. Either way, you can do solid amounts or percentages.
If you only want to do registration in the backend... you could create the events, but in DT Register unpublish it so it does not show up on the frontend for registration.
You can get to registration either through JEvents or DT Register. JEvents is actually not required to be seen on the frontend at all, unless you want the calendar view. Through JEvents, there would be no display of event capacity and such unless you put that info in the event details on the calendar. When someone clicks on the calendar event, there would be a registration button automatically embedded that would take the user to registration. If you go through DT Register, the user gets a nice looking list of events, sorted by category, and you can customize what data is shown there (location, capacity, number registered, date/time, price, etc. It includes a registration button, optional attendees list button, optional details button (linking to an article about the event), and shows notification if the event is full or closed.