Can a CSS class inherit one or more other classes?

There are tools like LESS, which allow you to compose CSS at a higher level of abstraction similar to what you describe.

Less calls these “Mixins”

Instead of

/* CSS */

#header {
  -moz-border-radius: 8px;
  -webkit-border-radius: 8px;
  border-radius: 8px;
}

#footer {
  -moz-border-radius: 8px;
  -webkit-border-radius: 8px;
  border-radius: 8px;
}

You could say

/* LESS */

.rounded_corners {
  -moz-border-radius: 8px;
  -webkit-border-radius: 8px;
  border-radius: 8px;
}

#header {
  .rounded_corners;
}

#footer {
  .rounded_corners;
}