Dmitri Gurov definitely is the protagonist of Chekhov's story, since the entire story is told from his point of view, the story begins with him, and he intiates the meeting and the love affair in Yalta. The only external conflict is in the facts that both principals are married and live in cities separated by a great distance. Both the man and the woman experience internal conflicts because, for one thing, they know they are doing something condemned by society and religion. They both think they should separate permanently, but they both are so strongly attached to each other that they cannot give each other up. The only other alternative would be for both of them to leave their spouses and live together. It would be very difficult if not impossible to get divorces in that time period in Russia. They would have to give up their children. Gurov might lose his job, and Anna would have no income. No doubt her husband would keep custody of her children, as Anna Karenina's husband did in Tolstoy's novel.
The setting is a summer resort. The countryside is beautiful but there is nothing at all to do. Everybody is bored. That is why the opening states that they were all talking about a new visitor on the promenade with a pet dog. They have little to talk about. Gurov initiates an affair with the married woman because he is bored and because he is a man with no inner resources--he doesn't meditate, or read, or commune with nature. He is an extravert as well as a womanizer.
Chekhov's story has been given many differeent titles by English translators, including "The Lady with the Pet Dog," "The Lady with the Dog," "The Lady with the Little Dog," "The Lady with the Lap Dog," "The Woman with the Pet Dog," etc., etc. I personally like the title "The Lady with the Pet Dog" best because it seems to suggest a little dog like a Pomeranian. The pet dog is an important character in the story because it brings the man and the woman together. It isn't hard, as Somerset Maugham wrote in one of his Ashenden stories, to get to know a person who owns a dog.
Joyce Carol Oates has further complicated matters by writing a story titled "The Lady with the Pet Dog," which retells Chekhov's story from the point of view of the woman called Anna rather than from that of Dmitri Gurov. A discussion of Oates's story is also to be found in eNotes, with questions and answers.