I bought it on Google Play store back when it was Dream City Underworld 
OK, if you get flu, it will take 48 hours to recover. To completely remove the flu, you need to sleep until full energy while recovery is 0. To check progress, you can refer to Statistic screen. There should be health progress under Health tab. This is the same flu mechanics like what we implemented from previous game. No idea why you didn't notice it before since you already have old game.
As for buying grocery, you can check out cake & pastry shop next to diner. It's actually baking supply shop so you can find flour, egg & all kind of baking stuff.Some players probably didn't notice the shop since it has no door from outside.