How do I verify the integrity of the sandbox JS, especially since I potentially have to download it each time I need it? Similarly, how do I verify the integrity of the asm.js code I'm receiving? Integrity is just as important, if not more important, than static analysis---for example, a malicious JS crypto library could pass a static analysis test, but intentionally generate weak keys.

There are ways to integrity checks manually, of course, but as far as I know the browser does not perform these integrity checks automatically (and it would be difficult to do so--you'd need to implement JS signing, and you'd need to implement PKI to get the right public keys to verify the JS signatures).