Yes, each of the kids will have a card. The cards will be tied to YOUR credit card. Initially, all cards have charge ability. If you wish to cancel this, you go to guest services. They will punch a hole in the card, which means "no charging."
As for spending on the ship, yes, everything goes thru the card. However, there is no need for a child to be able to spend money when you are not around. There is no chance to spend money in the kid programs (other than the teens, where they can buy smoothies or sodas). You can take your child to the gift shop or wherever else they might choose to spend money.
You can take your Disney Dollars or your Disney Visa Rewards Card to guest services and have this money applied to your account at any time during the cruise. You cannot "spend" Disney Dollars or reward dollars in any other way on board--only by applying them to the account at guest services.
Yes, each person's card can be tied to a different credit card. However, if nothing else is specified, each card is tied to the master card for that stateroom. That means that at check in (or later at guest services), you need to make sure that each card is going to the correct credit card. It's easy and only takes a minute to do this. IF you do not cancel your kids' charging, it is unlimited and tilts to your credit card.