I was wondering if I can write self closing elements like in XHTML in HTML5, for example, can be
HTML5 can either be coded as XHTML, or as HTML 4. It's flexible that way.
As to which is the correct way, that's a preference. I suspect that many web designers into standards are used to XHTML and will probably continue to code that way.
You can go straight to: http://html5.validator.nu/ to validate your code, or if you have the right doctype, the official W3C site will use it for you.