In A Farewell to Arms, is Catherine Barkley a round, dynamic, or static character?
There is a bit of confusion in your question in this respect. A round character can be either static or dynamic in a work of literature. A round character is a character who is well developed, but the character can either grow and change during the narrative (dynamic) or remain essentially the same (static).
Catherine Barkley has received much critical comment as one of Hemingway's fictional female characters. Critics usually view her as being a projection of male desire, a beautiful, loving woman who lives solely for her man, meeting his needs and complying with his wishes. Catherine loses herself completely in Lt. Henry; he becomes her "religion," she says. Other critics, however, find strength in Catherine even as she devotes herself to Fredric.
Since the novel is developed from Henry's first person point of view as the narrator, all we know of Catherine is what he tells us. His perceptions of her frame her character. She does not act independently from him. We know little of her personal history, except that when she meets Frederic she has suffered the loss of her fiance and feels unbalanced and emotionally vulnerable to Fredric's advances. Consequently, Catherine's character is not well developed in the novel. Her role is to complement Hemingway's protagonist and to advance the plot as Lt. Henry is developed as a Hemingway code hero during the course of the narrative. From her initial encounter with Henry until her death at the novel's conclusion, Catherine remains essentially a static character. Despite all that happens to her along the way, she remains essentially the same woman, Fredrick Henry's (and perhaps Hemingway's) romantic ideal.
A "round" character according to E.M. Forster in his "Aspects of the Novel" is a character who changes not just physically but also mentally and emotionally throughout the novel.
Catherine Barkley is certainly a "round" character who changes dynamically throughout the course of the novel.
Most modernist novels are existentialist and are characterised by the fragmentation of the identity of a single character. Hemingway reveals to us different facets of the character Catherine in the following manner:
1. At the end of Ch.5 Frederic tells Rinaldi that he and Catherine are "friends,"
So you make progress with Miss Barkley?
We are friends.
2. At the end of Ch. 14 they are lovers.
3. In Ch. 18 they are virtually husband and wife:Catherine says:
"We're really married. I couldn't be any more married."
4. In Ch. 23 Catherine feels like a whore :
"But it isn't nice to feel like one (a whore)."
5. At the end of Ch.23 Frederic quotes two lines from Marvell's "To His Coy Mistress,"
And always at my back I hear
Time's winged chariot hurrying near
Was Frederic hinting that Catherine was like a mistress to him.?
6. The novel ends with Catherine's death after she had given birth to a still born child. The physical changes are: when the novel begins she is just another woman but after she meets Frederic and falls in love with him she becomes an unwed mother who gives birth to a still born child.
So Catherine is a character with multiple facets to her personality and it is impossible to comprehend her fully: she is friend, lover, wife, whore, mistress and mother of a still born child.
More research would certainly reveal more complex facets to Catherine's multiple and dynamic personality and character.