Not necessarily. The hash could have been generated with something other than javascript.
In fact because functions like sha256 are iterative it's possible to hash a password which is longer than the RAM in a system. Technically possible to hash a password which is longer than storage in a system too, if you don't care about storing the password.
Probably not. The point of a cryptographic hash function is to be resistant to analysis.
Can we prove it has the much simpler property that toggling one bit of the input will, on average, toggle half of the bits in the output? (Probably not.)
If you calculate a billion sha256 hashes and look at the results you'll have an even enough distribution to say it's proven, but, it's not "mathematically" proven.
In fact because functions like sha256 are iterative it's possible to hash a password which is longer than the RAM in a system. Technically possible to hash a password which is longer than storage in a system too, if you don't care about storing the password.