can I use public pretrained word2vec, and continue train it for domain specific text?
fastText pretrained models should give you a boost to classification task.

gensim on the other hand has possibility to load model and train it with new texts but if you need to account for new words, you need to use


  build_vocab(update=True...)


So you can take with fastText pretrained embeddings to gensim and update with your texts.