How does a Child Theme works?

You are correct, that WordPress will load the style.css file from your parent theme first, and then the style.css from your child theme. These means because of how CSS itself works, you have to specifically override any rule you wish to override.

What this means is that you would need to modify the CSS within your Child Theme’s style.css to be:

.header {
   margin-top: 15px;
   margin-bottom: 10px;
   width: 100%;
   height: 150px;
   background: none;
   border-width: 0;
}