I love doing original character cosplay. The first group of cosplay I ever made to attend A-kon in 2004 was all original characters for the six of us. They had stories because they were from a book my friend and I were working on at the time.
As for answers to your questions:
Inspiration: I get inspired by so many things. Right now I'm working on a steampunk wind-up doll
inspired by Abney Park's song "Herr Drosselmeyer's Doll" and a yuki-onna
inspired by Japanese Folklore to fit with A-kon's theme for this year (we will have a whole group of yokai for the theme, each will be designed and made by me, mixing traditional Japanese looks with my own twists to them). I sometimes see a piece of fabric or jewelry and start thinking about what I can create with them. Other times I see a type of character or thing and want to make a costume for it. I've started working on costumes for the four seasons, Fall
being complete, and have often thought of doing costumes based on flowers (even though I'm so far from girlie most of the time) because you can do something beautiful with them. You just never know where inspiration can come from.
Back-story: Yes, some characters do have deep backstories. Others have just little bits and pieces here and there. Again, for my wind-up doll, I already have explanations for why she will have certain things or why stuff is going on with the outfit. It's just how my mind has developed the costume. Others might not get a lot of explanation, but they don't go without reason for things.
Why: Like others have said, why not? But I love the creative process. And as much as I love to make things based on already established series or art, I love to create my own things. I've been designing clothing for the last fifteen years and always like to bring my own designs to life along with the ones I do based on anime/manga/video games/movies...whatever. It's the thought of being able to find what I want to use and use it without anyone second guessing my choice. Creative, artist license. That's what I love about originals.