D.C.
Cats and dogs can be tough, because either they are good with kids, indifferent, scared, or really bad around kids. And they never change, there's next to nothing you can do to improve the situation with the cat. The cat will likely always be mean to your kids. Separation is the best option - baby gates, cat doors, keeping it in separate rooms is going to be the best option, unless you can find a new home for the kitty.
On a side note, see if your area has a volunteer cat care and shelter group. They may be able to help find another home for the cat.
On yet another side note regarding the poo outside the box. Is the box getting cleaned out on a regular basis (once a day or every other day) Cats can be fussy about where they go. It's not unusual for a house-broken cat to go outside the box if a litterbox is to full, if the box does not have enough litter, or if the box is obstruced in some way. Is the box in some place remotely "private." Cats can also go outside the box if they feel threatened or scared. Something to check.
Good Luck!