Mix-ins. Think Java 8 interfaces, with default implementations (so you can still have a putAll method as part of the interface, but the default handles the looping logic for you).
You can also do a layering-up type thing, where you have a base class that implements particular functionality, and inheriting from particular classes surfaces it or restricts it in particular ways.
Indeed. And once you have interfaces with default implementations, adding state gets you even more power. For example, intrusive reference counting and intrusive data structures generally are very useful applications of multiple inheritance.