Vaccinations protect the vaccinated person. But they also protect others (who would come into contact with the vaccinated person if she were to be infected) and the population at large (as the disease has less of a chance to spread if there a fewer usable carriers).
Given this situation, a classic collective action problem can arise. The cost and risk of getting vaccinated are mine alone, while much of the benefit is dispersed. So it is quite possible that it is better for each not to get vaccinated (regardless of what others do) even while it is also better for all that all get vaccinated than that none do. (In numbers: Suppose the cost and risk of getting vaccinated is 6 and the benefit to oneself is 3 and the benefit of another person getting vaccinated in a 101-person community is 0.05. Then, if all get vaccinated, everyone gains: 3 + 100*0.05 - 6 = +2. But each can reason this way: The first term is +3, the second is whatever it is independently of what I do, and the third term is -6. So I will lose 3 if I get vaccinated, whatever others do.) The sad outcome is that no one will choose to get the vaccination.
What can be done to overcome the problem? We can reciprocally agree to get the vaccination. But if this reciprocal agreement is voluntary (a "coalition of the willing"), then any individual has reason not to join for the reason canvassed above. So the only way to overcome the collective action problem is by agreeing, through a democratic decision, to enforce the vaccination on all . In that democratic vote, the two possible outcomes are yes (all are forced to get vaccinated and everyone is better off) or no (no one is forced and none or very few get vaccinated).
The best justification for allowing such votes with consequent coercion here is, in my view, that they offer the only way of overcoming the collective action problem. Letting people's individual rights be decisive here would frustrate the point of these rights: our protecting our interests. We fail to do this when we get stuck in a collective action problem.
About your second point. Yes, there are risks. Some may die as a consequence of getting vaccinated (albeit fewer than would have died without the vaccination program). How do we justify the program to them? We can say that the vacination was beneficial to them ex ante, that everyone's risk of premature death was diminished by the vaccination program (relative to its non-existence).
But there is a problem. Some may be known to be especially likely to die from being vaccinated. (For them the cost and risk may be 9 in the above numerical example, say, so they would be better off without the vaccination program.) I think these people should have the option to exempt themselves. Fortunately, they are generally only a few, and their non-vaccination therefore has little impact on the benefit the others achieve.
Your third question poses no special difficulty. If it's OK democratically to overrule the expressed preference of an adult with regard to her own body in this case, then it should be alright to override her preference with regard to her child's body in the same case. We can say to her that it is better for all children, including hers, that they all are compelled to get vaccinated than that their parents have discretion (and thus predictably veto the vaccination of their children). It's in the best interest of the children that we do not allow any parent to do what's in the interest of her own child.