By law i really don't think so. It does say on the actual game that M games are recommended 15+. Meaning that the child does not have to be 15 to play them. BUT he/she cannot buy the games. So it really depends on what you feel is right. :)
I think that actually letting them play would be a bit sad on the parents behalf. It is exposing them to violence and drugs and so forth that toddlers don't need to experience in their early years of childhood.
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Minecraft videos are for entertainment, how to play minecraft and etc.
I don't think it is ILLEGAL to play Private Servers. However, to own a Private Server is illegal.
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You can't. To play online you need an activated premium account, which costs money. It's also illegal to play with an unlicensed copy of Minecraft, and WikiAnswers will not provide guidance for illegal actions.
If it is the classic version that is free anyway on the minecraft.net site it is not illegal. if it is the proper minecraft that has survival and you download it free it is not illegal BUT it is illegal if you play it without a payed account. just because other people have it it does not make it illegal.