Is there a free (preferably open source) IDE you'd recommend that supports Python 3,

ActiveState Komodo Edit is free and open source, that also supports Python 3.

Cross platform (Windows and Linux)

Since Komodo is based on the Firefox code base, it runs with native look & feel on Windows, Linux and Mac OS X.

Standard features: syntax highlighting, auto-indent, etc.

Yup.

Run button, rather than console where you have to type "python foo.py".

Yup. Here's a blog post of mine demonstrating an use (using pep8) of the Run dialog. There is also a "Toolbox" feature when you can save Run configurations.

Contest-aware help (e.g. when they hover over math.floor() it should give help for the function).

Komodo does that (Cmd+hover also works), and more.

Embedded Python shell. Built-in debugger

These two features are only available in Komodo IDE. ActiveState provides free licenses to qualified educational institutions.

Useless crud stripped, e.g. they don't need refactoring tools during an intro class and auto-completion gets in the way of teaching.

Auto-complete and other features can be disabled in Preferences.

Along the same lines, it should be lightweight as some kids will go home to old PCs unable to run beefy IDEs such as Eclipse.

While Komodo is definitely not as beefy as Eclipse, it depends on what you mean by "old PCs". If it can run Firefox 3, then Komodo (which is based on Firefox 3 codebase) too should be able to run on it.