Word2Vec: Using pre-trained models
I think in a lot of cases, when people are using pre-trained models, they force their words to lower and singular case and then test to see if the lowercase and singular (not plural) version were trained. Sometimes, you have to check yourself like A.M. Bittlingmayer said. 

To provide some insight into why sometimes people remove stopwords versus not --- I think it has to do with whether they want to value infrequent words more or less. Because you don't want to value too highly very frequent words, the algorithm 'punishes' extremely frequent words. So, one could argue that because this already happens, you don't need to remove stopwords since they will be 'punished' anyways. So, not everyone removes stopwords.