Are LDA clusters identical across different runs? for a given corpus are the Latent Dirichlet Allocation clusters for it is unique in general?
How about the gensim  multi-process implementation of LDA? are there unique or they will be different for every run of code?
No, as there is randomness in the method implementation, for example here (in LdaModel of the gensim library). Hence, it can affect your final result in each run. Therefore, if you want to keep the result reproducible, you can set the random_state property of the model to a constant seed (see the documentation for more details).