You're correct that the NN-training aims to minimize its prediction-task errors over the course of training. However, neither the original word2vec.c code (upon which the gensim implementation is modeled) nor any of the other implementations I've seen make the errors for individual examples, or a full training epoch, easy to access.
The `neu1e` variable you've noted is the error as it back-propagates to the word-representation layer (`syn0`); the error on the output layer (predictions) is a few lines earlier in those methods, `ga` or `gb` (depending on HS/negative branches).
The `score_*_pair` and `score_sentence_*` methods may be of some use to your analysis, as they tally prediction-errors to determine how well new text examples match a trained model's expectations. Note, though, that they're currently only implemented for hierarchical-softmax mode.
- Gordon