What is wrong if your two year old Chipin dog is sensitive to touch on ear and legs and randomly screams in pain?

This could be any number of things, ranging from touch hypersensitivity to neuritis (inflammation of the nerves) to a learned behavior caused by being handled too roughly to an over-reaction from inadequate socialization.

I would start with a visit to your veterinarian to rule out biological problems - you'll want a complete physical and neurologic exam and a complete blood chemistry and differential cell count. This will probably run you several hundred dollars, but it will give your veterinarian a great deal of information about what the cause of the observed symptoms are. If there is neurologic inflammation (brain, spinal cord, peripheral nerves), any infection or a congenital defect that is pinching a nerve somewhere, this should find it.

If no biological basis for the observed signs is found, the next stop should be a referral to a veterinary behaviorist; I would suggest someone that is certified to do this and not just any dog trainer. A full behavior assessment as well as a detailed history of when your dog is screaming in pain, including what is going on at the time, will give your behaviorist a lot of information about the underlying cause of the behavior.

Although I would definitely want to rule out any biological problems, at this age and in this breed cross I would strongly suspect a behavioral problem rather than a biological problem. Small dogs in general, and designer cross-breeds in particular, seem to get a number of bad behavioral traits due to inadequate training and socialization as puppies; this often doesn't become a major problem until the dog is a year or two old.