Would Apple still take its 30% cut if I collect donations for charities in an app I develop?
I have here some App Store Review Guidelines, but you can't use in-app for donations. There are some points in the Apple In-App Purchase Policy covering some of your questions:

21. Charities and contributions
21.1 Apps that include the ability to make donations to recognized charitable organizations must be free
21.2 The collection of donations must be done via a web site in Safari or an SMS

Source:
https://developer.apple.com/appstore/resources/approval/guidelines.html (late 2013)