As for DD accessing the internet....
Well firstly my DD has been accessing the internet for many years before having a phone, and even if she didn't have a phone, I would be a very naive person if I thought that would stop my child seeing stuff online - to start with, all their friends have them where she could see it.
I very much believe in education rather than trying to bar access. I have taught my DD about online safety for several years and she is very sensible regarding this. She has even planned and taught lessons to younger children for a week on this very topic - she really does know what it is all about.
And even more importantly she knows what to do if things go wrong online. But this goes for everything - online and offline. I can't protect her from everything but I can do my best to prepare her with education and knowledge of how to minimise some risks and what to do if anything every goes wrong.
We have rules about Instagram - what type of images are acceptable, who she can be friends with (she must actually know them in real life), etc.
And I check her phone. She knows I do and that it is part of the agreement of her having a phone/tablet. I have all her passwords and our photo stream is linked, so I actually see every photo she takes. I don't do it daily and she is often there when I do it anyway. As I have said, she's had the phone 17 months and I still do it every so often.