Terminology: Documents are to be classified into 10 different classes which makes it a multi-class classification problem. Along with that if you want to classify documents with multiple labels then you can call it as multi-class multi-label classification.
For the issues which you are facing,
nltk.NaiveBayesClassifier() is a out-of-box multi-class classifier. So yes you can use this to solve this problem. As per the multi-labelled data, if your labels are a,b,c,d,e,f,g,h,i,j then you have to define label 'b' of a particular document as '0,1,0,0,0,0,0,0,0,0'.
Feature extraction is the hardest part of Classification (Machine learning). I recommend you to look into different algorithms to understand and select the one best suits for your data(without looking at your data, it is tough to recommend which algorithm/implementation to use)
There are many different libraries out there for classification. I personally used scikit-learn and i can say it was good out-of-box classifier.
Note: Using scikit-learn, i was able to achieve results within a week, given data set was huge and other setbacks.