I'm not sure myself. I thought the regular japanese thing there is that sneezing is caused by people talking (usually badly) about someone behind their back?
Even if you hand wave all the dialogue as translation issues/errors, the author tried hard to red herring it with Nana spending extra time with Kazuto right after he superhumanly outruns a car to save her from a molester.
Ijichi looks completely normal.
And by normal I mean fantastically suitable to that style that it doesn't look like theres anything different about her.
There was a microwave in the background of her with the burnt cookie and it made me think "did she microwave the cookie to bake it? of course it wouldn't turn out well..."
eh...I somehow got the vibe that the grandpa's time left in the world is limited and he wants to have slime be familiar enough to carry on if the old man can't solve it in time.