I live in Las Vegas and I have a tiny backyard. My mother and I really want to spruce up the backyard but can't do anything too elaborate (especially with how often the plants die out here). Also-- I will not put cactus in because my cats go into the backyard with me. I really want a Japanese theme but I have no idea where to start. Do you? (Any ideas at all appreciated)

First off, Cat's aren't stupid...lol, you can plant cactus, they will not go near it, and if by chance they do, they will only do it once.
I know that Las Vegas is a hot climate, so when planting I would suggest using a cloth ground cover over the soil to keep it from drying out to quickly. Then research online plants that are suitable for your area. Make sure you check the size of each plant when full grown to insure that you aren't over crowding your garden or planting tall stuff in front of shorter things. I live in NY so I am not sure what kinds of plants are good for your area. But draw a diagram of your area, Measure it and then you can make up a diagram of where each plant will go and how it should look when you're finished. Oh...also when checking plants, see which ones come back each year and which ones do not. That is a big factor when it comes to cost. Especially if you're on a budget and don't want to re-buy every year.
Good Luck and have fun.