Do I have to authorize purchased iOS apps to run on iOS devices? I can't seem to find a straight answer to this.

Do I have to authorize iOS apps I have purchased on my iOS devices to run on other iOS Devices? For example I have an app that I bought on my iPhone - can I sign into my friends iPad, re-download it, and then sign out (of the iTunes store) and use and update the perfectly? Or do I have to authorize it in some way?
Apple has worked hard to try to avoid the situation you mentioned. It is a major pain to have apps downloaded from different Apple IDs on one iOS device. For example, someone bought an app for me, but used (obviously) their Apple ID. Anytime I wanted to update that app, they had to enter their Apple ID in. That also meant that clicking "Update all" in the app store didn't really work. I finally got frustrated and bought the app myself.

So the short answer to both of your questions is basically yes. You can download it for a friend, but, anytime they update, you must autenticate for that app. Everything else should work normally.

You mentioned iPhone vs. iPad - as I'm sure you're already aware of, note that the app must be compatible for iPhone and iPad for what you mentioned to work.