In captivity the most common prey offered to monitors are crickets and mice. The larger species of monitors can easily take rats as well. Various other insects can also be offered. Hissing roaches are becoming more popular due partially to their large adult size and their prolific reproduction. Superworms, meal worms, silk worms, stick insects, fish, crustaceans and snails are also commonly offered.
Whenever possible one should offer whole prey rather than any processed diet. Dog food is offered by some people however it is my belief this is not a good food source. There is no roughage in dog food and this leads to runny stools. Another diet being experimented with is a ground turkey based diet. Again there is a problem with roughage.
This depends on many factors. A growing monitor should be feed everyday. For hatchlings I recommend feeding enough crickets or other insects, of appropriate sizes, until the monitor no longer shows interest eating. Then throw in a few more so it has something to eat if it becomes hungry later in the day. The size of the prey should be adjusted to the size of the monitor as it grows. Prey should not be wider than the width of the monitor's head. Smaller prey items are easier and more efficient for a monitor to digest. As your monitor begins to slow down in growth, feeding should also slow down so that the monitor does not become obese. Feedings can be every other day and the amount of food should be less.
Some argue that feeding mice can be harmful and cause impactions in the digestive tract of the monitor. Impactions only happen as a result of poor husbandry, not due to diet. Dehydration and inadequate temps are the causes of impactions. Monitors have very effective digestive systems and when they have access to proper heat (130 degrees +) their metabolism becomes high enough to digest anything. Proper hydration is also important for digestion. If the monitor becomes dehydrated its urates will become large and solidified which make them difficult to pass.
Most monitors will accept live or dead prey without hesitation. The easiest way to feed them would be to throw the prey into the cage. If you must jiggle pre-killed prey in front of your monitor then you should not do it by hand. The monitor can easily miss the food or mistake your hand for food. You should get a long pair of tongs and offer the food from them. Monitors also remember where their last meal came from. If it came from your hand then the next time you stick your hand in their cage they may think it has food and you will get bit.
This is really up to you. Insects should be offered live as there is no reason not too and monitors tend to ignore dead insects. However in my experience monitors will readily take pre-killed or live rodents. Live food offers a chance for your monitor to get exercise while trying to capture the prey. However you should not leave live prey in the cage for long periods. Mice can get hungry and start to bite your monitor. I have never dealt with this problem as my monitors eat their food very rapidly and if your monitor does not eat right away it might be a sign of improper husbandry.
Growing monitors need lots of calcium to support bone growth. A good quality source of calcium is needed while growing. The two brands I use are Repti-Cal and Miner-all. It is especially important to dust insects with a calcium supplement. Monitors on a mouse based diet do not need as much calcium supplementation as those on an insect based diet. However remember that pinkie and fuzzy mice are not a good source of calcium.