Dropdown menu, z-index problems

This will fix it for you:

#header_wrap {
  position: relative;
  z-index: 10;
}

You often need to set the z-index on the outermost parent (particularly in older versions of IE).

Just add position: relative, z-index won't work without position.

header ul {
    display: inline-block;
    float: right;
    height: 30px;
    z-index: 999;
    position: relative;
}