Why predicted proababilities from this binary classifier does not sum up to 1?
You are absolutely right to be skeptical about the results you are getting. I've been using boosted trees (not necessarily C5.0) for over 5 years, and I've never encountered results like what you've described here. Excluding the possibility that you simply had a typo in your code, I can think of two reasons:


Low decision threshold: As is often the case in class imbalance problems (e.g., fraud detection, marketing response, bankruptcy, etc) a low probability could still indicate the minority class if the decision threshold is also low. In your case, if the model might have automatically decided to pick a threshold of 0.05, then 0.1 would be flagged as C1.
Cost function: This is what you mentioned -- if your model has a high cost associated with missing true positives then 0.1 could be classified as C1.