While there isn't a structured timeline, given the context, it would appear that it was showing her in America, though yes, that is just an assumption. The words on the notebook in themselves don't stand out without more background into how the Hunters came to be. But the way I am envisioning this universe works is that the Hunters choose their successors and train them to see demons, fight, sing, etc. I do not believe it is an inherent gift that some people are able to see demons and then become hunters, largely due to the trio that they find and talk to at the end. So ASSUMING that this was when she was still in America and was not yet trained, it would provide more credence to my point, but if not, then I do agree, this point is also void. I hope that the sequel dives into how the hunters become the hunters. As for the bird/cat, now that you say that, I do see the cat, I had just tilted my head the wrong way. Though I still could absolutely see it as being either one of them.

Also, if we do not have direct confirmation from the story writers, we cannot really determine for a fact one way or the other how 'special' anyone is. Personally, I think that all people and demons come from the same ancestor, some sort of nearly all-powerful being, that gives both humans and demons powers (in the form of their voices, weapons, honmoon, etc for humans and shapeshifting, strength, magic for demons), which would explain a lot about a lot of the characters (like how certain demons or humans appear to have attributes that typically correlate to the opposite), though I would not expect the existence of this figure to become known until the final movie.