Does the Twitter app for OS X support URL minifying? I'd expect that when I paste a new link in the new tweet window the link would be auto minified or I'd have the option to minify... but this doesn't happen.

Am I doing something wrong or is this expected behaviour?
It's the expected behavior. Links are shortened after sending a tweet and only count as 20 characters when composing a tweet.

Twitter has shortened all links automatically with t.co since 2011. Many clients display t.co links with the target URLs, but you'll still be taken to a t.co URL first when opening a link. See Twitter Help Center | FAQs about Twitter's Link Service (http://t.co).