Although I've never had children (and quite probably never will), I have spent a great deal of time thinking about the approach I would take, particularly in the case of my brothers. The eldest is 11 years younger than me and I took a great deal of the childcare for the first 7 or 8 years. Basically until I moved out. I had a number of fundamental disagreements with the way my parents were raising the boys.
I view a major responsibility of any parent to be to enable a child to be independent. Not necessarily able to survive alone in the wilderness at age 3, but a parent needs to encourage children to learn the skills which will allow them to function when the parent is not around. For me, it includes things like teaching the child strategies for dealing with bullies where possible, rather than going to the school to complain as a first resort.
This also relates to a bringing up a child in a non-standard household. In order for a child to become independent, they would have to be able to think for themselves. Used correctly, a non-standard family or political structure can assist with this. I most certainly do not have a 'normal' life or set of attitudes but I do not think that this would be anything other than an advantage to child. I would hope to use it to explain the reasons I made the choices I have, which other choices are available and why someone might choose to take those other routes.
I do know people who have had their children removed from them due to their religious beliefs. They are practicing pagans and the social services have decided that this is synonymous with child abuse/devil worship and have, therefore, removed their children from their care. In one case, the girl was told whilst she was pregnant that they were planning to take the child off her as soon as she'd given birth!!! To my mind, this interference is abhorrent. I don't believe that the state should have this power. I don't actually believe that this would stand up in court if the person concerned could have afforded a top lawyer, but this sort of thing rarely happens to those who are that well off.
In answer to the specific questions then:
1. I would argue that a parent should raise a child to question everything. This is one of my central beliefs about life though, so maybe I am saying that they should be raised according to one’s beliefs. I think this might get circular.
2. People have the right to doubt your skill at anything for any reason at all, or even for no reason. They just don't have the right to do anything about it. They can think what they like, but they have no right to be listened to or have their opinions carry any weight at all. And you have the right to be pissed off at them for telling you about their opinions of your skills.
3. No. There should be no difference at all between raising a child in a Christian household and raising them in a communist household (is that supposed to be capitalised? I never worked that one out), or indeed a polyamorous household, or a bisexual household, or a pagan one, or, indeed, one in which all the family are members of the Conservative party. The same rights and responsibilities apply.
Also, I should make it clear that when I'm talking about a child being independent, I'm not advocating the, kind of situation that elphaba2 described, with the child making all the decisions for the family. Young children are not just smaller adults and should not be treated as such. One important thing to learn is that there are some decisions that have nothing to do with you and that you don't get a say in those. What colour car the family has would fall into that category for me. I also wouldn't say that there should be a distinct point at which a child goes from making very few decisions to making many. The ability to make a decision should be linked to taking responsibility for the outcome and should increase gradually throughout life. Taking the example of “What should we have for dinner”, I would advocate linking that with a responsibility to cook. So a 9 year old could be told that they cook on Tuesdays (with assistance where needed, obviously) but that they get to choose the menu (within reason). If they make an unhealthy choice, talk through with them why that’s not a good idea every week, but sometimes it can be allowed. Rights should be linked in to responsibilities.
It might be worth noting here that I really do believe in children having chores. I had to do half of the cleaning of the house from about 9, as well as all my own laundry, some of the cooking and almost all the tea-making. My brothers (15 and 14) don’t have chores yet (except keeping their rooms clean-ish). I firmly believe that I got the better end of that deal. I keep a clean and tidy home and was able to fend for myself from very early on. They aren’t trusted to boil the kettle (possibly rightly so). It all comes down (for me) to equipping a child for life.