4 years later: I have run into a similar problem where I want to train a multi-label classification model using lightgbm/ xgboost. To the best of my knowledge, there still isn't one for lightgbm but you can implement it on xgboost: https://xgboost.readthedocs.io/en/stable/tutorials/multioutput.html

So the answer to your question 1 is yes, for lightgbm the labels have to be mutually exclusive.