Most kittens receive a hostile reception from other household pets, especially from another cat or animal.  The other cat (animal) usually see no need for a kitten in the household and these feelings are reinforced if it perceives that special favoritism is being shown the kitten.  The existing animal must not feel that it is necessary to compete for food or attention.  The new kitten should have it's own food and food bowl and should not be permitted to eat from the other cat's bowl.  Although it is natural to spend time holding and cuddling the kitten, the existing animal will quickly sense that it is being neglected.  The new kitten needs lots of love and attention, but the existing cat (animal) should not be slighted.   In fact, the transition will be smoother if the existing animal is given more attention than normal.  The introduction period will usually last a couple of weeks.

The existing animal will remain hostile to the kitten.  Fighting may occasionally occur, especially if both try to eat out of the same bowl at the same time.  This is an unlikely occurrence if competition for food and affection are minimized during the first few weeks.

The existing animal will only tolerate the kitten at first. The existing kitten will hiss at the newcomer for a day or two, then the hostility will cease, but the existing cat will act as if the kitten is not present.  This is more likely if the existing cat is very independent, has been an only cat for several years. 
I takes awhile until the kittens to decide to play, and sleep near each other.  This is more likely to occur if competition is minimized and if the existing cat has been lonely for companionship.

When you bring a new kitten home, I suggest placing the kitten in a small area, like a laundry room or bathroom the first couple of days, so the kitten can get used to the new people, smells, noise, etc of the household.  Only you will know when it's time to give the kitten a little larger space to explore.   Let them know where the litter box is, then let
the kitten explore on their own, they will ignore the other cat for a few weeks before being comfortable together.

We always suggest having 1 more litter box than the number of cats that you have in the home.  If they have litter box accidents in weird places, that occurs out of the blue a couple of times, it could be that they have a bladder infection and should be tested out by your vet.  Folds like clean litter boxes, I scoop every day, as they will go somewhere else if the box is dirty.  If that does not solve the problem, change the brand of litter.   Some cats have a preference on what feels best on their feet.  I find the litter at a feed store is usually better than what you buy from your Walmart.

If the cat wants to scratch something not allowed, rub Downey sheets all over it to discourage this behavior.  Pet stores sell products for this also. 

If a cat is hyper, or anxious, we rub lavender scent on their cheeks.

A great website that answers a lot of your questions is