How important is "recognition" to you? (regarding going as original characters)

Going with an original look or character concept is highly admirable, but you have to remember - few people can come up with individual characters, stories and images on their own. Creative talent is uncommon, and even creative people can be derivative, making amazing replicas but not constructing a completely personal creation.
So, the "what are you supposed to be?" question is understandable. Most folks don't think outside certain boundaries and don't expect other people to. They expect others to do what they would do, which at Halloween is dressing as an identifiable figure.

It starts early, too. Kids are asked "What are you going to be this Halloween?" and the answer is usually a bat/cat/zombie/witch/ghost/ninja/cowboy/princess/alien/superhero.

Or, they are asked WHO they are going to be, which is usually answered with (for guys) Batman/The Joker/Darth Vader/Capt. Jack Sparrow/Jason/Leatherface/Indiana Jones/The Grim Reaper/Dracula, etc.

Mardi Gras doesn't have this issue. The individual vision, the unexpected costume, is part and parcel to the entire carnivale atmosphere, so the look is the important thing, not the identity.

I think you'll find that most adults will tend to appreciate an original costume moreso than kids, even though some of them may ask who you are. Even if they don't know a character or have no vested interest in the story of that character, a well-crafted and visually arresting costume will meet with high approval and may even inspire others to think outside the box.

Anyone can buy a costume. People stop and appreciate handmade and original work.
Same thing goes for props and decoration.
And yes, it can be so jarring for people that they may try to find a cozy label for it, and ask "who are you supposed to be?"
Tell them you are something of your own design, and leave it at that.

With personal experience behind my comment, I have to say you will get all those responses (and probably others). I've been a reptilian sea monster-type creature, a theropod dinosaur skeleton, and a dragon-faced maiden. My Lobo-drake costume (which is in my avitar at the time of this posting) was just a costume I built around a mask design that had no connection with an existing character. I love bones and in particular skulls, and that costume was a chance to play with that and jump into techniques that were totally alien to me. Last year's Undead Bastet, while based on the cat-headed Egyptian goddess, was a costume that I put my own spin on (like Lainie mentioned) and again was a grand experiment. All of these costumes have been met with similar comments as you listed, as well as "do you make those masks to sell" and my favorite responses: silent fear, nervous giggling or a gasp .

This year I'm going as Poe's raven "Nevermore," and this one is also very much a blend of what comes out of my head and little bits of imagery harkening to my favorite poems and stories by Poe. This will be the first time that I can fairly confidently say what I'm supposed to be, but since I am not literally a recognizable character that anyone really has pictured in their mind as I do, there will still be that little hint of uncertainly and questioning in the face of those asking. Many people need nice contained categories, and they may have a bit of trouble bending their minds around your own unique design. Still, while they are gradually understanding that very few things in this world fit into perfect little identification boxes, others will appreciate the originality. I really feel that unique costumes provide mistique and wonder to Halloween costuming that would be sorely lacking if everyone was a pirate, witch, and vampire (though we need those, too). Variety truly is the spice of life, and that goes for costumes as well.

Yes, the "what are you supposed to be?" annoyed the heck out of me at first, but you really do get used to it and I've decided that, since I myself don't always have a neat and tidy answer, I keep the ambiguity going and let them decide what my costume represents. The growled answers of "your worst nightmare," "that which isn't," "that which follows you in the shadows," and "what lurks below" are some of my favorite ambiguous answers I'll give to the "what are you" question. Even annoyances like that can become a chance to have a little fun.

I like it when certain people ask me "who are you supposed to be?" because that is often asked by the people that have taken no creative thought in their costume. It means I will be appreciated by the right demographic that put alot of time and creativity in their costumes. I think character research is important if you intend to incorporate dialogue as that character and part of it is finding some creative responses in character to questions like "who are you supposed to be?" or "wow, how did you make that?"

I try to make my costumes as creepy as possible so I care about response more than recognition.

This will be my first halloween with my sister and two nephews and I remember asking my little crumbsnatchers who they wanted to be for halloween. One wanted to be the joker and the other wanted to be harry potter. When they watched me construct my stalkabout I could see the kids put their thinking caps on and their creative minds started wondering. Now they are making a costume from scratch.