About Nozomi not being able to wear a ring at work - when I first read the strip, two things immediately came to my mind:
She might be working somewhere with a dress code. Many Japanese businesses/companies have a strict dress code which you are expected to adhere to. I know some more conservative companies don't like their female staff wearing pants and high heels. They are also discouraged from wearing heavy make up or jewelry.
She doesn't want to openly flaunt her relationship with Eli because Japan isn't exactly very open to gay relationships.
I thought she might be working in a food related job, as a chef, apprentice, kitchen assistant...
Here in Japan they are very strict if you work with food. That means no rings, no bracelets, no nail polish, nails always cut short and clean... if you cut your hand you can't even use normal band aid, you have to use approved liquid bandage. The supervisor checks everyone's hands before starting work...
Also no facial piercing (like eyebrow piercing, nose piercing, lip piercing, etc...), no earrings and no necklaces (jewelry in general) because the risk of them somehow getting in the food.
And no make up. Because the risk of the make up products getting in the food.