How to use binary relevance for multi-label text classification?
Just some views/suggestions. After removing the stop words did you stem/lemmatize the text you got?  That would probably reduce the number of unique words in your corpus and brings some forms of words to the same level. But caution while using stemming as it sometimes create noise. 
Try postagging and see what can be the important tags you want to keep and eliminate the ones that you feel are giving less relevance to the text.
Did you try to find the some important terms from each document by using tf-idf or chi-square method. They might be helpful to see the relevance of terms for each class/document.
See how far can you reduce the dimensions of the matrix after using the above, if not you have already used them and then apply logistic regression or whatever classifier you wanted to.


On your second question.


I don't think having data from different sources should be a problem since you are trying to create model based on keywords, Hopefully it should be able to see. I am not confident on this part.


I had worked on text classification before with bigger documents and larger corpus with not so fruitful results with various models.

Hope this helps.