I agree, there's something fishy about those instruments and the fact that Hwata is supposed to be an old man. Yet, in this Goryo era, he is a legend. His coming from the Heaven was supposed to have taken place a long long time before the Goryo Era so it could be that as time went by, the Lady from the Heaven became the old man from the Heaven... But if this diary and those tools are real hers, how come she doesn't have any memory of this ? Does coming back to her own time imply that she'll forget everything about her time-travel ?
About the ending, I am also a bit worried. I don't know how it could end well if he doesn't go to the future. But if he goes there, then he will not die as it's written in history... Or maybe, he'll die in his time, she'll go back to her time and meet his descendant or something...