How word2vec can handle unseen / new words to bypass this for new classifications?
One way to do this is to use context information to represent each word along with the w2v vector. You can choose to represent this information in any way you like: add another 600 dimensions (100D w2v vectors for 3 left and 3 right context words), just another 100D as the sum of context vectors or any other fixed length representation for your context.

When you're training, you can use a version of 'word dropout' that will utilize this information. 20% of the time, set your w2v vectors to zero, forcing your classifier to use the context dimensions to represent the word.

When you encounter a new word, the hope is, the classifier learned to use the context information as well as it learned to use the w2v information.