What is the difference between Jest and enzyme?

Jest is a framework which includes a task runner, assertion library, and mocking support. This means it can execute different unit test cases, write its result in console or log files, create mocks, or verify all the assertions. In short, it will execute the test.

Enzyme, on other hand, is a library that provides a simple interface for writing unit tests. For this purpose, it wraps packages such as React TestUtils, JSDOM and CheerIO. React TestUtils has methods to render a React component into a document and to simulate an event. JSDOM is a JavaScript implementation of the DOM (Document object model). The DOM represents the tree structure of UI components. CheerIO implements a subset of jQuery core and is used to query the DOM.

Enzyme is not a test runner. It doesn't have its own assertion library. It just provides a collection of APIs for unit testing. That's why it could be integrated with Jest or any other task runner.

Yes, you can use karma with jasmine.

Aaron's comment answers your first question.

Enzyme provides unit testing utility functions for React components, such as allowing shallow rendering. The Enzyme docs say you can use it with any test runner or assertion library - see http://airbnb.io/enzyme/.

Yes, you can use karma with jasmine.

As far as the best way to test a react-redux project, that's a separate question and I suggest you search Stack Overflow. :)