Use a new parser for new pages, the old parser for old pages. The introduction of H.264 didn't suddenly render all older formats unplayable. Why can't a new markup format coexist with older formats, too?
It’s possible to do it but it’s also needlessly complex while giving you very little in return. The biggest, most problematic flaw of allowing browsers to parse invalid code, namely that different browsers might handle failure differently, is in the process of being fixed and that’s good enough, I think.
Strict parsing has been tried and it was a resounding failure. It’s not going to happen.
2. What would the point be, apart from annoying every single end user?