Hello Joshua,
Thanks for writing in!
We have built all the features and design as optimized as possible to display perfectly in most modern browsers. What we do not have any control is the version of the browsers. In some older browsers, there might be conflicts or issues which is due to how the browser is rendering the html or handles the css of the page. Old browsers do not support html5 and CSS 3 technologies.
1.) There is no way you can view the page across different browsers in Cornerstone. You only have the responsive screen previews and not with the browsers. You will have to go through testing between browsers, Chrome, IE, Firefox, Safari, etc. manually.
2.) You can use page emulator sites and check your site to figure out what is not displaying correctly in what browser. Perhaps this can help:
Hope this helps.