1) No, there are many different kinds of jobs that chemical engineers do. Some, like sales or research and development, do not require them to operate plants or work with machines.
2) Typically in a lab setting
3) No, typically not.
Since you are a sophomore, I think it is better to just transfer to another school and shoot for an engineering degree. If your interest is in Chemical engineering, you will need additional courses in chemistry, in addition to math/physics.