Do you believe in Karma?
Not the way most people seem to, no. Definitely not in any way similar to the Wiccan "Threefold Law."
Do you believe we carry karmic debt from one life to the next?
No. I believe that we get our "oh shit's and "attaboy's" during the life in which they are earned.
How would one "clear out" negative karmic debt, if that is possible?
By making restitution to the people you have wronged. Being nice to "Alice" sometime down the road isn't going to make up for the wrong you committed against "Jill." Giving money to charity isn't going to make up for money stolen from those who rightfully earned it. And so on...
If you do not believe in karma, do you believe in something similar?
I believe that "karma" is a self imposed reward and punishment system... and one that apparently for most focuses on "punishment." Basically, we create our own punishments when we knowingly do something wrong (this, of course is assuming that the "wrong" we have committed is a moral wrong and not a legal one). There is a notion of "crashing by design" that I'm particularly fond of. When we do something wrong enough to feel guilty over it, be that something done deliberately or accident, we will eventually set ourselves up to be "punished" for it. We will "crash by design..." our own design. Why? Because only by being "punished" can we be relieved of our guilt, especially if there is no way to make amends to those we have wronged.
Now, since someone usually brings up the question about "bad things" happening to good people for no apparent reason, I still don't believe that "karma" has anything to do with it. Nor do I believe that terrible things happen as some part an unknowable, divine, or "higher" purpose. I believe that sometimes bad things happen to good people simply because "shit happens." (Of course this should not be confused with the occasional whacks from the Great Cosmic 2x4 some deities take such delight in swinging our way when we've done really monumentally stupid.)
The only "lesson" I believe we need to learn is that while we may have little to no control over some things in our lives, we always have 100% control over how we choose to deal with it. We always have a choice, even if that choice is to submit or do nothing. I also believe that our eventual happiness with ourselves and our lives is completely in our own hands.