Which browsers should work with the site?
Any modern browser (i.e. released in 2015 or later) should work: Firefox, Chrome, Safari, Edge etc. They need to have support for HTML 5, CSS 3 and Javascript.
Should all the pages here validate as HTML5/CSS?
They should, yes. However, validation services at W3C (http://validator.w3.org/, http://jigsaw.w3.org/css-validator/) are pieces of software running behind W3C web servers. As such, they change from time to time and become more strict. On occasion, what validates today won't validate tomorrow. If you find such pages, let me know.