Did some html in my eclipse today because I needed to do something on the web with Java. Never thought HTML could be that complicated when you want zero warnings in your consule..
Generally all syntax checkers for html/css should be disregarded. There is really no standardisation for it between browsers, and indeed if you are trying to create a responsive design, then almost no syntax checker will know what the hell its meant to be checking css wise.
No standard user will be checking the console, either. And those users that ARE checking the console will probably understand the difficulties of cross-browser compatibility and know that the warnings aren't going to eat them
Warnings are better than it not working. It's what happens with different browsers, it's the way it is.