Does the output of the Sequence-to-Sequence encoder model exist in the same semantic space as the inputs (Word2vec)?
No, assuming your input vectors are one-hot encodings. These input one-hot encodings are in an $n$-dimensional Euclidean vector space. The last hidden layer of an LSTM is not due to the non-linear activation functions across the encoder. Therefore, an average of the inputs will not necessarily align well in a vector space with the model output, nor are you guaranteed any similarity in cosine/Euclidean distance.