In short, yes. In kantei a sword is judged by the workmanship, the workmanship confirms the "mei (nakago)" and not the other way around. But, as we can see from O suriage papered swords there is a wide range of possibilities from almost definitely, yes, this person made this sword, all the way to a general school/tradition attribution. The better the sword the more likely the answer will be more specific.
see previous answer.
Then there will always remain questions about what exactly was the original nakago. One adage from a collecting point of view says "half the value of a sword is in the nakago."
The sword would (almost undoubtedly) get bounced at shinsa. A trained professional could refinish the nakago, but that's a whole other conversation in itself.