If you're feeding chicken it has to be plain with no seasoning. I normally use a George Foreman grill when I'm doing chicken so there's no oil on it at all. I'm not sure about coconut oil, I don't think it would be bad for hamsters since I know a little bit of desiccated coconut is fine. If you fry the chicken first with no salt or pepper on it then take a bit out for her then that might be okay. I don't know for sure so maybe wait and see if anyone else has a more detailed response
If you're allergic to eggs I wouldn't bother buying them in just for her, I eat a lot of eggs and normally just give my syrian the top of a boiled egg, about a cm square. If she's getting mealworms she doesn't need eggs for protein (some people argue they're not great for protein anyway) so it would be a bit of a waste for you to buy a box of eggs just for her. If she likes mealworms you can stick with those, you can buy packets of the freeze dried ones or I found
these which have mealworms, crickets and shrimp. My syrian doesn't like them unfortunately but if your girl likes the dried mealworms they might be great for her.
As for safe foods I generally use the list on the
Hamster Central wiki. I generally just give her whatever I have in the fridge myself I don't really buy anything in specially so I guess brocolli, spinach, green beans, carrots, cauliflower, bell peppers, cucumber and sometimes banana if my mum can spare a bit of hers. She also gets a few grains of cooked rice or a bit of pasta, some tuna if it's in springwater, chicken or boiled egg white. I do buy her peanuts specially.