1) Do you consider it indoctrinating children when parents teach thier children certain belief systems?I've answered these in reverse order when I typed them. My initial response to this was going to be "Yes, obviously", but after answering the other two questions below, I think it is more likely to be - "sometimes, it depends".
i.e. it depends on what the belief systems are.
Literally, indoctrinating someone is to give them, or force them into, doctrine. And doctrine is not so much the basic beliefs of a religious system (e.g. there is or is not a god or gods; Jesus died for our sins) or even the literal words of a particular holy book (be that the Bible or "The God Delusion"). To my mind, doctrine is more the interpreted meaning of a belief system as it applies to the real world we live in.
I don't know if this is a real division, but in my head there is a difference between the basic core beliefs or theology of a particular religion, and the doctrines of that religion.
So teaching children about the theology of a faith - I will use Christianity as an example here as it's the one I'm most familiar with - such as "there is a God, He loves you, He has a Son called Jesus who died so that everyone might be able to go to Heaven. We should kill or steal or hurt one another deliberately or for no reason, because Jesus told us to love God more than anything and to love everyone else as much as we love ourselves. And we shouldn't judge other people because that's God's job." - the real broad-brush strokes that all denominations of Christianity would agree about. That's fine. At worst, it's harmless (at best - the bits I've picked - probably quite a good thing), because it doesn't teach them to hate or fear anyone else whose beliefs or lifestyles might be different from theirs.
Doctrine, on the other hand, is more problematic, because to my mind it's doctrines, and this faith or denomination's claims to be more doctrinally pure and "right" than other faiths or denominations, that lead to all the bad aspects of relgion - the divisiveness, the hate, the bloodshed. I'm good because I have the right doctrines; you're bad because you don't. Or - worse, IMO - you're bad because my doctrine says you are or that the things you do are bad.
BTW I have no real problem with "you're doing things and you should because my doctrine says they're bad" and no problem at all with properly-constituted democratically-passed laws that outlaw this or that behaviour. Man's laws important too, and those ones we should - by and large, indoctrinate. "Don't murder people, or you'll go to jail" carries more weight in my mind than "Don't murder people or you'll go to hell". (But that's my mind.)
2) Is it morally, ethically, or legally inappropriate to "incodrinate" your children?"Incodrinate"? I'm sure that the child welfare people would have something to say about inserting large deep sea coldwater fishes into your children. (Sorry for the typo humour

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In the USA and all of the EU I know about, there is no law against "indoctrinating" children in a particlar belief system. So (occasionally regrettably) it is not legally inappropriate to do so.
Morally and ethically? Well, it's a quality control issue, and the best way to deal with poor doctrine is to argue it out. Kids should have the opportunity to discuss their beliefs in school, and the other kids (and the teachers) should have the opportunity to quiz them about it. And, if necessary, rip their stupid ideas to shreds*. If mom and dad don't like that, they can always either teach their kids to think more rigorously or learn to do it themselves.
* And it's the stupid
ideas that should be examined and, if needs be, ripped to shreds, not the kids themselves, their parents, family or friends, either literally or figuratively. Very smart people can (and often do) believe in very stupid things, and it's only a culture of intellectual challenge that will keep everyone on their toes. This is sadly lacking almost everywhere today, in the pressure to only learn stuff that either is useful for future employers (as in schools) or superficially funny or entertaining (as in most of the media). (Kudos to

for having the idea of thrashing out ideas as the main reason for existing.)
Or, there's always the back door of "faith schools" or even home skule-ing. Not meaining to be too pejorative there - not all home-schooled people are badly taught, but I would go out on a limb and say that most of the really bigoted, stupid or ignorant attitudes extant today are not taught in schools but in homes and churches (of all shapes and sizes).
3) Are belief systems such as religion harmful?They can be. Even if it isn't you doing the harming - ask a Holocaust survivor.
And, obviously enough, belief systems (all kinds) can be twisted to allow them to justify terrible things. You can have a pacifist atheist like, say, Bertrand Russell, and another like Chairman Mao who's responsible for millions of murders (I chose those two because I'm a functional atheist myself and I didn't want any believers to think I was getting at their particular belief system, however crazy or dangerous I might privately think it was).
People criticise activist Christians as being right-wing social authoritarians, and that MIGHT just be true of the noisiest activist Christians in the USA. But Desmond Tutu is an activist Christian on anyone's definition, and is about as far away from the first type as you can get, not only in politics but in religious beliefs too, while remaining a member of the same religion.
The Dalai Lama is everyone's favourite poster-boy for Buddhism, and most Westerners I've spoken too elevate him - and Hindu gurus too - to saintly status (perhaps rightly for the Dalai Lama, he does come across as a very good and moral man) while conveniently forgetting that the chaos of bloodshed and the invention of suicide bombings comes from an essentially Hindu vs Buddhist conflict in Sri Lanka.
Morons and murderers are evenly distributed through all belief systems (including systems of non-belief like atheism).
Ultimately I'd like to live in a world where kids are educated - by families, by schools, by the media, and by society as a whole - to think for themselves, and to take very little on faith alone. They should certainly learn
about religion - about the good and bad points of all the major faiths. It certainly shouldn't be kept off-limits as a topic.
I think the best thing would be if religion were treated like sex in terms of child development.
Not talking about it at all is stupid, because kids will come up with their own ideas from playground gossip and, possibly, some shady characters. And introducing the subject in too much detail too young is, frankly, a bit creepy, if not outright abusive.
But, it's obviously a very human subject and - for most people - it's an essential part of their daily lives. It's absolutely critical for how we came to be here (in the sense of how our society came to be the way it is, rather than how we came into being at all - I don't want to derail this into an argument about creationism). And it's a very important subject for discussion, education and debate.
But it's one that is, at root, a private matter of individual conscience, where unless anybody gets involuntarily hurt and everybody consents, pretty much anything goes.