How a web application works across different browsers and how it is actually developed in reality so that it can works across different browsers has major differences. Do not overcompensate code for a feature that can use a small amount of code. The HTML used in your application will be described by the Doctype. One of the best things to do is validate your CSS and HTML so that problems are avoided in the future.
The best way to make a responsive website which is faster and easier to build is to get to know the features first. To make sure your browsers are following the same rules and their behavior is consistent, you can the CSS reset stylesheet. Style sheets can be linked to different browsers using conditional comments. To know more about the tips on building cross-platform compatible websites in detail, read the article here.