Yes. For not yet understood reasons, children do not seem to develop symptoms from covid19, but they can still be carriers. Combined with the fact that schools are essentially giant petri-dishes for disease, and old people are particuarly vulnerable to covid19, means that sending a child to grandma is essentially playing Russian roulette with her life.

I would go so far as to suggest that if she can take care of herself, grandma should self quarantine as much as possible for the duration of the crisis. People do not become less dangerous just because they are relatives.

After the family spent 2 weeks in self quarantine, visiting grandma is probably fine.

At least in the US it is relatively uncommon to live with extended family like grandparents. If the children are already infected, then yes, a change in proximity to grandparents would increase the risk of transmission. If schools are closed before many children become carriers, then no, simply having grandparents near kids probably doesn't change much for the grandparents' risk.

Yes, in fact they are.

Specifically, younger people are sometimes being told to avoid older people, sometimes when the older person is in need.

IIRC, some place had bans/restrictions on visiting nursing homes because the potential for spreading the virus might end up exposing a large percentage of the occupants to a significant chance of death.

No citation because I cannot remember where I read it.