"what age should children be resposible for their online behavior?"
That's up to their parents to decide. Legally, the parent should be just as responsible as they would be for the offline behaviour of a child under their care.
"What role should parents play?"
Everything, as far as what they want their children to access or not. No parent should complain to anyone about how their children are "able to" or are "let" do X online. It's the parent's responsibility to ensure their child is educated (well, hopefully school can help out) about the internet, and to take whatever measures they need, such as installing filtering software. Some help may be warranted for education on installing filtering software, and its limitations.
"Are they aware of the breadth and type of content available?"
Everyone knows there are things on the internet a parent may not want their children to see or do. Claimed ignorance about this is just an excuse for lazy parenting and passing on the blame.
Children will access the Internet at their friends' places, on their friends' mobiles, at the local library etc unless your family is a part of some restrictive cult. I imagine that to protect themselves legally up to the children's age of 16 and while they are still are at home, parents would need to show they've made good efforts. Education is key and doing the basics to protect them where you can. I.e. that might be home PC-based filtering solution up to a certain age. Outside of the home society needs to help parents in this respect. It's all very well to say all parents should know about the Internet and not claim ignorance but we're in a transition stage here and will be for many years to come. Some 19% of people don't have access to the Internet in NZ, so some understanding of their situation is needed.
Kids using the internet is a bit like letting them go outside to play, you can't always be there, but you hope they have the responsibility to do the right thing. Unfortunately this environment is much larger than your average backyard, and holds a lot of nasty stuff we don't wont our children to just come across. Parents must lay down some important ground rules for Internet use: Number one being, never have a computer in the kids bedroom! It is in the lounge or somewhere where some other adult is always present. If no one is there then the parents much ensure only certain trusted sites can be accessed. This next one is unpopular: Let your children know in advance that all their internet use will be checked - daily! This means that mum dad or big bro or siss will check on all the sites kids have been to - I can hear the boos.
I think a lot of parents want to be able to control their children's access to some extent, but most play the computer illiterate card or just cant be bothered looking up the correct methods to do so. I myself have pass codes on EVERYTHING I own. Before you can use my iPod, Laptop, Cellphone or even connect to my internet router in my room, you need my password or PIN. I know all my parents passwords, but they know and I know, that they will never know mine. Most of my friends are the same as well. They have passwords on everything mainly to stop their parents snooping around. I'm not going to lie and say I haven't seen anything I shouldn't have on the internet, but I know people who have seen stuff at a way younger age than I did. I saw my first porn popup at 10 but by that time my mate had shown me his dads magazine collection. Kids growing up in this day and age will always be one step ahead of their parents. Ask any teenager 16-18 if they'd leave their laptop in the family room, open, turned on and logged in with their parents sitting in the room while the teen is away. I know no one that wouldn't lock their computer before leaving it, it just become a force of habit.
The only way I can see that parents can get a hold of this, is if they install something where the line enters the house. But even then us teens will still find a way around. Just like sneaking out to go to a party. We'll always find a way.
Late 2013 MacBook Pro with Retina Display (4GB/2.4GHz i5/128GB SSD) - HP DV6 (8GB/2.8GHz i7/120GB SSD + 750GB HDD) iPhone 6S + (64GB/Gold/Vodafone NZ) - Xperia Z C6603 (16GB/White/Spark NZ)