Not quite. You just store an id indicating which stylesheet to load. When loading the page you would then check for the id in the cookie, and if the cookie does not exits use the default style. You...

Apparently when using z-index you are meant to specify an absolute position, but it also works with a relative position. Maybe try:
#logo{
position: absolute;
width: 100%;
height: 80px;...

Yes, but AFAIK it would require a little scripting. You could save the style in a cookie, then select a stylesheet based on the value when the page loads.