They have for years, but the browser was historically very stripped down and didn't get get certificate or SSL/TLS algorithm updates, leaving parts of the web inaccessible as time went on. I only ever use it to hit Project Gutenberg for public-domain ebooks (though their redesign a few years ago made that a lot harder)

Yes every kindle since the very first one had an "experimental" web browser in various stages of usability. All currently supported kindles have a web browser that is fairly usable though very slow.