As I’ve mentioned briefly in a previous ask (http://bills-pokedex.tumblr.com/post/166442752080/um-hi-yes-my-houndoom-hannah-has-recently-begun), for the most part, caring for newborns is complicated, but it’s made easier by the fact that unless the mother has rejected her kits (unlikely, but it happens), she will generally care for her young. This is very true for the eevee family, which are generally protective of their young due to how much the base species is affected by its environment. Likely, your flareon will display similar behavior towards her shinx young, so you won’t need to worry about things such as food or basic care until they grow old enough to be fully independent from their parents.
The same could actually be said for your predicament with your shinx kits. While it may be frustrating to see them unleash their electricity seemingly at random, their luxray parent will instinctually attempt to train them himself unless he’s rejected his young. All you’ll need to do is make sure your shinx kits are given ample space … and that you safely store any object you’d miss.
Once they’re old enough to be weaned, caring for the young is very much the same as caring for the parents, only on a smaller scale, so if you remember how you cared for your flareon and luxray when they were eevee and shinx, you should have few problems. If, however, you had received these pokémon as evolutions or if you don’t remember at all, simply scale back the amount of food and water you give the kits and train them for less time, but provide the same basic care and scale upwards as the kits get older.
If, however, your flareon or your luxray have outright rejected your new kits, feed them formula from a bottle at least once every three hours. (As you get closer to weaning them, you can increase the time between feedings.) Be sure that the formula is warm (but not scalding) and feed them by holding them upright in the crook of your arm, not on their backs like you would a human baby. Swaddle your shinx in rubber-lined cloth (available at pokémarts) or be sure to wear rubber as you bottle-feed them. At one to two months, eevee and shinx can be weaned onto solid food. (Start with wet food in a shallow dish, then move on to dry food soaked in milk and then dry food without milk once their teeth grow in.) At this point, it’s important to keep an eye on their droppings to be sure they’re digesting their food properly. Luckily, eevee can be housebroken fairly quickly using verbal explanations. Shinx may need to be litter trained, however. Rub a cotton ball on each kit’s lower belly immediately after feeding to stimulate their digestive systems, then place them in the litter box and essentially … let them go. Eventually, they’ll get the idea.
Moreover, keep your kits warm at all times. Their ideal body temperatures at this point in time is roughly 100 degrees Fahrenheit. Add a hot water bottle to their carrier or bed if you live someplace cold, and monitor their temperatures regularly. Additionally, bathe your kits regularly by wiping them thoroughly but gently with a warm, wet cloth. Eventually, eevee may need regular baths with eevee shampoo (some, especially those that eventually evolve into espeon, will learn to bathe themselves), but shinx will need to be encouraged to bathe themselves.
Best of luck, and congratulations on your new additions!