Yes, url shorteners pass link juice But they need to be 'proper' 301's redirects using URL shortener's which does include Bitly and many others. Bit.ly does have great tracking analysis but I view them as far from stable. Clickbank links are often rendered useless and the fact that they have a .ly extension makes them far from safe.
Normally they do, but it depends on which URL shortener you use. ;
For example goo.gl uses a 301 re-direct and thereby is passing link juice.
URL shorteners are definitely useful when it comes to making your URL look prettier, especially, as we mentioned above, when sharing your content on social media. But they can be also useful when you are producing other types of content like a print ad or an infographic and you want to insert your link in a nice way.