Yes, stay on property, not just because it's Disney, but because the Disney complex is huge, it'll take you at least 20 minutes from any of the off property hotels to get to the parking lots of any of the parks. Plus, with staying on property you get to go to certain parks earlier or stay later on specific days.
Get the park hopper pass. That way you can go to more than one park in a day. You might not think this is necessary, but it can be very useful. Example, Animal Kingdom closes at 6:30 or so, still plenty of time to go to another park, especially one that is staying open later for on property guests.
GET THE DINING PLAN!!!! I cannot stress this enough. Like any other amusement park/sporting event/etc., the cost of food is outrageous. If you can get it pre-paid in advance, do it. Besides, it is totally worth the cost. You can do the all counter service option. But, I would recommend the next step up, which gives you 1 counter service and 1 table service meal per day + snacks. They have some really cool places to dine. We tried all sorts of food the last time we went. We had the buffets at Norway (which was a Princess Character Dining) and Germany. We used 2 table service credits per person to dine at the "fancy" African restaurant at Animal Kingdom Lodge. We wanted to go to the Science Fiction Dinner Theater at MGM, but could not get reservations. Counter service alone you can plan on $10 per person per meal, that's for a burger, fries, and a drink. For table service, the entre alone usually runs around $20 per person, and that doesn't include an appetizer, dessert, or drink.
If you do the dining plan with table service (or even if you don't and want to dine in one) make your reservations in advance. You can possibly get a reservation, but it may not be where and when you want. YOU WANT TO PLAN YOUR DAYS IN ADVANCE ANYWAY, NOT GO THERE WITH NO PLAN OF ATTACK. So, plan you dining as well.
As far as budget per day, if you buy the whole package in advance, you don't actually need a lot for anything else. Though, of course, you will spend money on other things. A lot of it depends on how much you anticipate spending on souvenirs. I would suggest giving each child a set amount of money to purchase their chosen souvenir(s). Believe me, every park or store you go in, you and they will see something that you just can't live without.
BIGGEST SUGGESTION, get the photo card from the first photographer you see. You know how parks have people stationed at entrances and such taking pictures, you get a ticket, and go see them later to purchase. Disney has this card you get instead. Then each time you see one of these people at the different entrances, autograph signings, rides, etc., you give them the card and they note your photo. When you get back home you have a month to look at the pictures, edit them, and purchase what you want. Or you can buy the disk which gives you the copyrights to them.