Can Japanese voice actors watch an anime episode for free after an anime episode is finished because they did contribute their efforts in it?
Voice Actors get paid. That is their compensation. And per se, voice actors aren't cheap. To specifically give them only the episodes they appeared in seems to be too much of a hassle. Just think how many side roles there can be in an episode.
That aside, occasionally special voice actors (main roles or special guests) can get gifts from the company they work for. E.g. the German voice actor Gronkh received a "LEGO Batman" Joker plushie for his work of speaking the Joker in the German Translation of the movie, ADDITIONALLY to his payment.
OP asks if watching the episode for free is something NORMAL Japanese voice actor get FOR their work. I say no, because Japanese have a special attitude towards their work. Being allowed to work for your company is something to be honored in Japan, almost as important as getting the money. If a company is especially grateful towards an actor, they may bestow him with additional honor. But that would be something of emotional value, not an episode of their own production everyone can buy. (At most, something like the first item of a collectors edition, and that's more of a gift for the producer. Voice actors would get something tied to their person or the role they spoke.) In addition Japanese are very polite, so it seems out of place to ask for free episodes if one really wants some.
And finally, it is work. Who has worked for a time, knows it is not always your wish to revisit your past work.