Reducing text input size into word2vec without affecting performance too badly?
I don't think there really is a right or wrong answer to the "removing stopwords" question.  Some people will argue that throwing away information will reduce model performance, while others argue that it'll increase noise.

I personally follow a simple rule of thumb.  If my model depends on sentence structure, then I keep stopwords.  If i'm modeling topics and more interested in important phrases, then I remove them.  This seems to work well.

If you're looking for ways to reduce you matrix, then yes removing stopwords is a perfectly acceptable idea.  Another thing you can do is apply common feature reduction techniques like LSA or chi2 to find the most important words and reduce your input space to the most meaningful words.  However, doing this may dramatically effect the performance or your word2vec model.  But if it is your only choice, then why not give it a go.