Does the moderation team sometimes look at asked questions and edit them so that they seem nicer? because a friend and I have been experimenting with what you'll let be answered and what you won't and I asked him a mean question, which got changed into a slightly nicer one shortly afterwards
No, the moderators don't change or edit questions.
If I turn off anonymous questions, does it un-check mark the option so they are forced to send publicly or does the question just not show up until you allow anonymous questions again?
If you turn off anonymous questions, the option is struck through and unavailable on your profile. It's not possible to send anonymous questions to your profile once you choose that option.
You can't do that from the apps, but you can go to the website and on any answer click the time stamp (like "2 minutes ago") which will show you just that answer. Use this URL to link directly to this answer.
Ask.fm says I need to sign up in order to ask an "anonymous" question. But before It was never required for me to do this...I could always ask questions without "signing up". I am curious does this mean I was reported for the questions I asked and now I need to signup or something?
No, we changed that for everybody to increase safety and overall user experience.