It comes with jsDom baked into it for browser behaviour.
Using a full headless browser to test with is not ideal, however I trust that you have a good reason to want that.
If that is a requirement for you, you can tie jest up to karma :), but I beseech you to have a very good reason to test against a headless browser first :)
Depends on what you want to test, I guess. If you want to test UI widget behaviour (not just business logic), then jsDom probably isn't the right tool and you need a browser. E.g. if you need CSS measurements or more tricky DOM behaviour.
It's not entirely clear to me why you'd want a headless browser though. Once you're already running a browser, you might as well run a full one, and also get the debugging capabilities.
Not out of the box, but the environment setup is easily configurable. I've been using a Protractor for my E2E tests since it also uses Jasmine... does the trick until there is more first-party support.