She sounds like a typical chinese!
I wouldn`t feed carrot too often though as carrot contains natural sugar, but a very small piece (about 5mm in size) won`t do much harm if fed in moderation. Mung beans (bought from the supermarket and left to soak in water for 48 hours) swell up and sprout a little tail. These are also know as bean sprouts when they grow long or are forced on. The water needs changing every day though as they can go gooey if left un-rinsed. They just need enough water to cover the beans. Hamsters love a fresh sprouted mung bean as a treat every few days.
I soak them and sprout them for my syrian and robo and they love them.
Handling: You are right not to force her into this. I took my time with prinny and when she was used to taking little bits of millet from my palm, I would wait until she was facing me and gently `cup` both hands around her and she would either jump off, or stay on my hands until she had a little kiss and then I would place her straight down again, seed treat at the ready as a training tool! Patience and time are the key.