While a parent should have his/her own life too, the foremost life obligation of a parent is to guide their child into successful, self contained and responsible independent adult life, which is a mix of protecting and sheltering the child while guiding the child thru the steps of independence and adult life. I think MANY parents come to see their child(ren) only as mostly annoyances they have to contend with and give very little thought about training and leading their children towards becoming an adult.
We are highly attentive but also "strict" parents. There are rules, the sanctions for breaking the rules is known and certain, there are duties relevant to the age and doing so required, but also it is a guiding process to advance the child too - which can then mean forcing the child to deal with realities of social life on his/her own - monitoring all the way - again age relevant.
My lady is the Pied Piper for infants and little children. I have never seen anyone who likes or is as liked by children and we often have many here. But the rules apply to them too (with some limits since she's not the parent). For older children, she is quite strict and uncompromising. At first they protest but most come to respect and even appreciate the attention. Most children are starved for supervising parental attention when young - instead just parked at a TV and video games - and by teens come to only see their parents then as obsticles - and goods and services providers.
IF the mother had not been outside watching, I would have no problem with the arrest because of the 6 year old.
Of the other matters, no reason to go round and round with you about it. I said my peace on it.
Good grief! At 6 years old I was building tennis ball cannons and playing Superman by jumping off the garage roof....and so were all the other kids in the neighborhood! Playing in the street was among the LEAST dangerous stuff we did.