In the anime, if a trainer battles with Zorua/Zoroark, will illusion activate making it become another Pokemon?
Yes they do. For example, in the movie Zoroark: Master of Illusions, Zoroark transforms into the three legendary dogs, Suicune, Entei, and Raikou. Zorua in the episode, Movie Time! Zorua in "The Legend of the Pokémon Knight"!, Zorua transforms in a female Meowth and even into Iris, a human.