Ibsen is considered the Father of Modern Drama. He is considered the first modern realist because he dealt with contemporary issues, I.e. he explores the role of women in modern society in The Doll's House. Nora could be considered the first feminist when she declares her freedom at the end of the play.
As for naturalist, I'm not sure what you mean by this. Admittedly realism and naturalism are closely related but naturalism is considered more gritty, I.e. The Lower Depths.
Although he may use some symbols, as do many writers, I would not consider him a symbolist. I see more symbolism in The Glass Menagerie.
By defying accepted practices in the theatre and writing about real issues confronting recognizable people, he opened the door to modern drama, thus the title Father of Modern Drama when referring to Ibsen.
Realist, as the characters and events are life-like and are not affected by supernatural and the like.
Naturalist, for he writes about characters and natures being influenced by heredity and past (the naturalists are highly influenced by Darwinian theory of evolution). Naturalism is a deeper kind of a realism and is concerned with individualism as such, which Ibsen too seems to be concerned about.
Symbolist, yes absolutely, as can be seen in his plays such as Hedda Gabler, where the setting itself is symbolic of the atmosphere. There is this inner room which symbolizes Hedda's past, there are those withered stately powers that symbolize the lost Glory of her aristocratic lineage, and there are the pistols that enhance her phallic character.