Phaser is an open source project and available to download for free. We use github to manage the project and it's on the github site that you will find the latest version:
https://github.com/photonstorm/phaser
Yes. Github provide the option to download the whole repository as a zip file. You can then unzip this locally to get to all the files. However we would strongly recommend you learn to use git instead. It will allow you to easily update to the latest versions as we release them, and it makes contributing towards the project much easier as well. If you don't fancy learning to use git via the command-line there are some great applications you can use instead. We'd recommend SmartGIT on Windows or Git Tower on OS X.
The Phaser repository is split across 2 main branches: master and dev. The master branch contains the most recent stable release and is the one you should try first. The dev branch is where we prepare the latest versions. In here you'll find experimental features and API changes, some of which may not work. But it's also where you'll see the bleeding-edge features we're working on.
Right now there is an approximate delay of around 4 weeks between dev and master. This means we build the new features out in dev and let them bake for a month before we move them to master. This gives them time for others to test and comment on them, and it keeps the release rate of Phaser to a manageable monthly cycle.