I have a div that has background:transparent, along with border. Underneath this div, I have more elements.
Curre
Yes, you CAN do this.
Using pointer-events: none along with CSS conditional statements for IE11 (does not work in IE10 or below), you can get a cross browser compatible solution for this problem.
Using AlphaImageLoader, you can even put transparent .PNG/.GIFs in the overlay div and have clicks flow through to elements underneath.
CSS:
pointer-events: none;
background: url('your_transparent.png');
IE11 conditional:
filter:progid:DXImageTransform.Microsoft.AlphaImageLoader(src='your_transparent.png', sizingMethod='scale');
background: none !important;
Here is a basic example page with all the code.