Can a CSS class inherit one or more other classes?

You can add multiple classes to a single DOM element, e.g.

<div class="firstClass secondClass thirdclass fourthclass"></div>

Rules given in later classes (or which are more specific) override. So the fourthclass in that example kind of prevails.

Inheritance is not part of the CSS standard.

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;
}