my best advice is that the 2 cats need 2 separate litter boxes. i know in my house that would never be very good because we have a small house, but ive read that this is a must for a home with 2 cats, the outside the box accidents might stop if they have their own boxes, or a free box when one is in use...
ive never used a "self cleaning" box, they seem .. too good to be true... and ive noticed that the peed on litter just squashes back up, so its no substitute for regular cleaning.
i have a litter box that comes with 2 bottom buckets, a strainer type bottom, and a top with a door. i found the door would trap my cat in the box because of the 2 part bottom... but i think if you had only one bottom, it should be fine, but our litter box is in our laundry room, where our son is never allowed (via a gate)
besides, its very unhygenic to have a litter box in your food prep area .... ewe. even ... just for airborn issues. LOL
what i would recommend is that everyone has a bathroom, or at the least, a tub. put the litter box in the tub or bathroom... so its out of the way, and the bathroom isnt a spot people normally let their children play.... so its safe from little hands.
they do sell mats to put right outside the litter box door, and it collects most of the runaway litter. the cover will protect litter on 3 sides from getting out... but the front will still have litter coming out from cats feet.... so those mats are a good idea.
another best bet is to just keep cleaning it, removing chunks once a day even. then it doesnt get to be such a big hassle when you do have to clean it. also, cats will tend to poo anywhere when the litter box isnt cleaned often enough! so... the pooing outside the box might also stop if you clean the box more often...
anyway, good luck with this. i hate the litter box. even though my husband wanted this cat just as much as i did, he "blames" us having the cat on me so that i have to clean it... and i know if i wait for him to do it it will never happen.... so i guess it dosent matter who does it as long as someone dose it. see if you can recruit some help from your teenage son, even if its just for a while... while your daughter is so young. but anyway..
again, 2 litter boxes, and frequent cleaning, covered but not self cleaning litter box... and remove the door most probably as it might trap your cat in as it did to mine...