1) Either of the first two work. Both "'My dream is to become a nurse," and "'My dream is becoming a nurse," are correct. 2) Options two and three work, but the third option is best. "What I like to do the most is play video games.'' (Also note that it's "video games" not "videogame.") However, this sentence is kind of unnatural in itself, so I would recommend saying, "My favorite thing to do is play video games." 3) Either of the first two. "Your duty is to defend the Wall from the White Walkers," or "Your duty is defending Wall from the White Walkers.'' Hope this helps!