Thank youf or trusting us with your question. I have a few clarifications I'd like to get from you so that I may better assist you.
Could you explain what you mean by "cut" is she snipping it with her fingers as if they are scissors?
How long has she been doing this for?
Have you asked her not to? What happens if she does not?
Thank you for the clarification.
More than likely the cutting is just a comfort tool. Many children develop little activities that they like to do that give them comfort. I'd not worry about it, it's quite possible that it has just become a habit too. I would suggest you keep an eye on her though, and it if seems to become extreme (for example if she refused to drink or eat something that took two hands because she couldn't "cut"), then you might mention it to her doctor and just have him evaluate her or refer her for evaluation. This shouldn't be necessary right now though from what you are describing.
As far as being an extremely picky eater, that is very normal for a 4 year old, and absolutely nothing to worry about. Most children start outgrowing the pickiness with their food right around the time they start school.