Your neighbor has every right to take pictures of people engaged in activity that she thinks violates the law or the HOA rules. That she is a lunatic doesn't make a difference, and I imagine neither the police nor the board pay much attention to her.
The threats are another matter. If she is making threats that are likely to cause serious emotional distress, she is engaging in harassment. You do not say what she is threatening. Is it to call the police about their non-existent criminal activity or is it to inflict violence? Being a pain in the neck is not harassment. Threatening bodily harm is.
Generally, unless the HOA establishes rules that say the kids cannot play in the common areas, then it is okay for them to do so. Common areas are for the use of everyone in the HOA.
Taking pictures of the kids in the common areas does not require parental consent nor is it any kind of actionable conduct. Sending the photos to the police or the board is also not anything giving rise to any kind of claim.
If the neighbor is threatening the kids with harm or injury, that could be considered a crime. If that happens, you could report that to the police, but beyond that I don't see anything that would result in your getting an injunction.