How word embedding work for word similarity?
I think you have it mostly correct. 

Word embeddings can be summed up by: A word is known by the company it keeps. You either predict the word given the context or vice versa. In either case similarity of word vectors is similarity in terms of replaceability. i.e. if two words are similar one could replace the other in the same context. Note that this means that "hot" and "cold" are (or might be) similar within this context.

If you want to use word embeddings for a similarity measure of tweets there are a couple approaches you can take. One is to compute paragraph vectors (AKA doc2vec) on the corpus, treating each tweet as a separate document. (There are good examples of running doc2vec on Gensim on the web.) An alternate approach is to AVERAGE the individual word vectors from within each tweet, thus representing each document as an average of its word2vec vectors. There are a number of other issues involved in optimizing similarity on tweet text (normalizing text, etc) but that is a different topic.