AKA Jodi's highly personalized, severely opinionated riff on romantic and fictional character arcs and static characters.
How necessary/important is it to show the hero growing? Does he have to grow? Also, if the hero has to grow/change too, does it have to be the same depth as the heroine? Can his growth be on a smaller scale?
Yes and no, and I'm sorry if that's one hell of an answer. In other words, it depends on what you're trying to do. In stuff like Mills and Boon and Harlequin Presents, when you're king of the heap and top-dog, there isn't much room for growth because that type of story isn't about the man getting in touch with his inner self, but the woman growing to love the man for who he is--a fully realized guy, with everything in place before the story starts. There is a character arc on the woman's part, but it's in response to outside or internal events. In other words, it's not a journey where two people meet in the middle, but a single journey, where the hero is the end-goal. It isn't necessary to show the hero's growth, but it is necessary to show the growth of his emotional bond.
ie. Say the hero is a rich Greek Tycoon (funny how that got capitalized, lol)--as the heroine moves forward in "her" character arc, the hero needs to show a corresponding emotional growth arc which is revealed (in the heroine's pov) as a softening or vulnerability. Because the heroine is getting closer, and the hero--while not changing--is falling in love with her. It's a growth of the relationship.
I consider this story to be more hers than theirs. The majority of the scenes are in her POV. In the few that will be in his POV, he's not interacting with the heroine. The scenes serve to reveal info about him and sometimes the other character in the scene.
The other place where this kind of thing works is fiction. Fiction is a huge umbrella. If the story is not primarily about the growth of two people toward each other, and not where the woman starts out in the one-down position, then it's leaning toward commercial fiction. I would suggest, and it's just a suggestion, writing the male pov scenes and later switching them over into the heroine's pov to keep it consistent. The same kind of info can come from the heroine's observations and serves the same purpose because the story is primarily about the heroine and her journey. Not necessarily a love story, but a story where love is a subplots. How you can tell if it's a subplot or an integral part of the story is to take out all the hero-related stuff, and see if there's still a story. If what's left is tangled and makes no sense, then the love story is a plot thread and deserves a little more attention. One thing that would help both is to make sure whatever is forcing the hero and heroine to interact is something so strong they can't escape. A mystery? Some kind of action-adventure?
It can be something as simple as having children who are friends, or dogs who are mortal enemies. Or maybe one person's dad is going out with the other person's mother. Or maybe one person is a spy and the other is a terrorist. Or maybe they're just drawn to each other, or the heroine wants the hero so much, her character arc and what forces them together is her realization that she does want him.
Is there a "rule" that all major characters have to have a character arc (is that the right term?)?
Yes, it's the right term. Another way to put it would simply to say, "Do people have to change?" Some people think it's a rule, but rules are made to be broken. Imho, it also depends on how short your story is, and I'm not talking length, but time. How much actual real-time passes in your story? Unless the story is about someone else, maybe a mystery--or how someone overcomes something, and the heroine is a great part of it, then if the story has a short time frame, the focus should be on the heroine anyway. The heroine is going through the story--the others, no matter how major they are, can remain static unless they need to change because of something in the plot.