Of course not, but if your button pushes/replaces history then it should be an <a> tag with an href property.

They should for accessibility reasons. If they're not assistive tools might not know they're even click-able.

Maybe if you're doing like a painting app or something you don't need to.

Button elements and other clickable elements are all detected by accessibility tools as such. If links were the only thing they could click, the modern web would be almost completely unusable, which isn't the case.