Certain foods and lots of contact with water can cause dermatitis. You need to look after your skin whatever you are doing in the kitchen – this includes handling and preparing food, as well as cleaning.
Yes you can. Once you've identified what's causing the dermatitis, you can take steps to prevent exposure and protect yourself. The only exception is if you have developed an extreme allergic response, especially to a food, in which case you may not be able to continue to work in a kitchen.
Yes you can. You just have to use one of the many moisturisers that are free of fragrance or taint and nut oil. The moisturisers should be in dispensers (not pots) to avoid cross contamination between different users.
If you are following good food safety practice, then you are also well on your way to preventing dermatitis. Remember you have to comply with health and safety law just as well as you do with food safety law.