When a child lives with both the parent and the grandparent, either one of them (but not both) may claim the child as a dependent and get all the benefits. You may decide between you who will claim the child (see exception below*). If you can't agree, then the parent has first claim.
You cannot claim head of household (HoH), whether you have a dependent or not, because you do not pay more than half the household expenses. Grandparent can not claim HoH, unless she has, at least, one dependent, even though she provides most of the household support. So, if you claim yourself and all the kids, nobody in the home can claim HoH.
*If the parents of a child can claim the child as a qualifying child but no parent so claims the child, no one else can claim the child as a qualifying child unless that person's adjusted gross income (AGI) is higher than the highest AGI of any of the child's parents who can claim the child (grandma must have more income than you to claim your child)
