Does the short story contain a climax/resolution? Explain the answer by using examples from the story. Why does James include these conventional short story elements?

First, you have to uncover what is happening. The most likely scenario is you are reading about a man who is emasculated by his wife. She is constantly harping on him about forgetting things:

'Remember to get those overshoes while I’m having my hair done,' she said. 'I don’t need overshoes,' said Mitty. She put her mirror back into her bag. 'We’ve been all through that,' she said, getting out of the car. 'You’re not a young man any longer.'

At the same time, you are left wondering if he isn't suffering from onset Alzheimer's given his age and her saying "I wish you’d let Dr. Renshaw look you over."
Regardless if he utilizes escapism to masculinize himself or if he is doing it because his mind is wandering, each mini-story in and of itself has a climax. Unfortunately, there is no resolution as reality breaks through and the reader doesn't finish the experience playing out in Mitty's mind.
If the reader pays attention to the story happening between his imaginative scenarios, one sees an old man, forgetful in nature, worried he will forget the item that his wife wanted him to remember. He knows that she will berate him for his forgetfulness and the reader is left rooting for him. When he does remember the dog biscuits, his wife is still not satisfied as he sits in a chair that she could not readily see him in. He finally, albeit weakly, takes a stand against her berating when he finally states, "I was thinking,” said Walter Mitty, “does it ever occur to you that I am sometimes thinking?" At this point, the climax is that he finally stands up to his wife and advocates for himself. The resolution is that she contends he is ill, probably because he never acts like this, she states "I’m going to take your temperature when I get you home." He knows he is bested and no matter what he says or does, he just cannot win with her, so off he goes back into the recesses of his mind and faces the firing squad. He has attempted to stand up to her and was shot down.