Washbowl
You need a washbowl when taking baths because you put your soap/shampoo/etc. in your washbowl, and take it to the shared public bath. Since you wash your body/hair outside the bathtub in Japanese baths, you can use the washbowl to scoop hot water out of the bathtub, and slosh it over your body to rinse your body/hair (although usually there’s a row of faucets/showers that you sit in front of, and you can either shower off your soap, or put hot water in your washbowl to slosh it over your body). You also take your washbowl to the Washing Room when you wash your face in the morning.
Red bean jelly
Normal red bean jelly comes in bar form, and you slice it into pieces. The jelly Eimi is holding is the type sold in convenience stores, and they are already in individual slices, wrapped and ready to eat.