How do we protect children from risk while allowing them to grow up (and without life grinding to a halt)? For eg, is it okay to leave a sleeping baby with a baby monitor while you pop next door for dinner, or to let the kids walk back from school aged 6, as they do in some other countries? How about letting 13 year-olds go off camping alone?
I think the main way you teach them about risk by doing stuff with them - walking/hiking, camping, visiting wild areas and big cities, navigating public transport systems, reading maps, road safety and you talk to them about 'stranger danger' (for want of a better phrase), informed consent, grooming, sex and relationships, and calculating the risks of any given situation. Children who have a good relationship with their parents and feel they can talk to them about anything, without fear of judgment are much less likely to come to harm than those who don't.
As for your examples, IMO small children should never be left in a house alone unless you're in the garden or putting something in the dustbin. Baby monitors are not fail-safe and neither are locked doors, cots that DC aren't supposed to climb out of, and pets that might get into areas they shouldn't be in. Plus, there is always the risk of fire and you're more likely to be able to get your DC out of the house safely if you're in the house to start with.
Walking to school age 6? Not round here. Too much traffic and two busy roads to cross. Plus six-year-olds cannot accurately judge the speed of approaching vehicles.
Camping alone at 13? Again, that would be a 'no' from me. They're too young to be able to cope if something went wrong - if something caught fire, for instance, or someone got stung and went into anaphylactic shock. Better to go with a group of their peers and a few adults IMO.
How does society view mothers who take physical risks in the careers or sports they pursue? Are there - and should there be - different 'rules' for mothers and fathers?
I think society is waaaay more judgmental of mothers than it is of fathers. Dad is a racing driver? No problem. Mum is a mountaineer? How selfish of her to put herself in harm's way when she has children at home. There is a complete double standard. Being a parent though would make me think twice about putting myself at risk. Before I had kids I only had myself to worry about, but now I'm responsible for them and their lives would be very difficult if I was to be maimed or die, so I need to factor that into any decisions I make with regard to my personal safety.
Alternatively, have you ever had to weigh up big, life-changing gains against similarly life-changing losses? Or taken a big risk on someone or something which paid off - or went disastrously wrong?
No, I haven't.