Yes chinchillas can eat carrots.
My chinchilla ate carrots and loved carrots when I was a kid.
I would feed my chinchilla a carrot a lot of times and the chinchilla enjoyed it.
Carrots are very healthy for chinchillas and most chinchillas do love to eat carrots and other vegetables.
Some chinchillas cuddle with people while others may not.
I've had a chinchilla that loved to be held and would cuddle with me but they are not the most cuddly pets unlike a cat or puppy.
But over all most chinchillas are friendly and some do love to cuddle.
Some chinchillas are friendly pets and some chinchillas may not be friendly.
It's all up to the individual chinchilla as to whether or not they will be friendly to people.
But for most of the chinchillas they are indeed friendly and I've mostly known chinchillas to be friendly at least the ones I have been around.
I used to have a chinchilla as a pet as a kid and it was really friendly and liked to be held and petted but I've been around a few other chinchillas that hated being held and petted.
However it can sometimes take awhile for the chinchilla to get used to you and become friendly if it will.
So if you do have a chinchilla and it is not all that friendly at first then give it time and it may become friendlier to you.