Agreed, giving a discount or other reward for recovering an abandoned cart is a VERY dangerous road to go down IMO.
I am not sure I agree. Many major online retailers do this and are very successful. Promotions can be site-wide or category sales, individual items, certain brands, offering mail-in-rebates, etc.
Done right, it is wildly successful.
Depends on your business and the core reason why carts are abandoned. This example was just demonstrating what a business could potentially do with push notifications; most businesses already do similar things on the web, to reasonable success.