how important is character
Question: How important is character in a story? Are the names of the characters in your story important?
There is no story without character, even in cases where the characters are non-human.
You can look at character two ways. First, to take a purely objective approach, Dramatica defines a story as a model of a mind attempting to solve a problem or rebalance an inequity. Characters are the elements within the storymind that represent the motivations and drives. Without them, there is no recognition of a problem to solve, and no need to solve it. Therefore, no story.
To take a more subjective approach, we read stories because we enjoy the insight into other people and how they deal with problems, because we too face problems in our lives.
It's part of our nature as human beings to learn from other people's experiences. The stories told to us by parents, teachers, friends, etc. become the inspiration for our own actions. They are the models we draw upon when making decisions.
Fictional stories are an attempt to impart wisdom in a more carefully crafted and condensed form.
But what makes a story work - what gives it the ring of truth - is character.
Throughout history, people have always wrestled with
how a person's character, his personality, affects the decisions and actions he takes and how it affects his life and - if he is a leader or person of rank - how it affects society as a whole. Stories, whether fiction or nonfiction, are a way to explore this issue.
In addition, readers also enjoy the interaction of characters with different personalities who represent different perspectives. There is much drama in relationships and fiction can take the place of village gossip.
As for your second question... At the very least character names are useful as identifiers to keep the various characters distinct in the reader's mind. Some authors like to create names that also say something about a character - whether it is their ethnic background, their family connections, or their personality. Some writers like to find names whose traditional meanings correspond to the characters' personalities. Other writers (e.g. Charles Dickens) like to give characters ironic names that sound like their personality. When writing a story, it's your choice how much meaning to assign to character names.
If you only have a few characters, you might get away with referring to them as "the man," "the woman," etc. if you want to be vague. But with multiple characters, names generally work better.