Would Apple still take its 30% cut if I collect donations for charities in an app I develop? I know that selling things without going through App Store In-App Purchases etc. is not allowed widely. Even apps like Dropbox historically had problems with it and even they use in-app purchases for Dropbox upgrades today.

If I'd develop an app that people donate charities using an online payment system (like Balanced or Stripe), would Apple still take my app down because I'm not going through in-app puchases?
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)