For household with children, kids should be taught how to move toward the pup from the side and how to hold a puppy. Young children should do this while sitting down to prevent a wiggly pup from accidentally being dropped.
It should be noted that trauma (including being dropped or stepped on) is the 2nd leading cause of death for the Chihuahua breed. This is an 'under the foot' dog that needs to be carefully watched and taken care of.
3) An Injury or Medical Problem.
While you did state that the veterinarian performed a checkup and did not find any medical issues, we must still point out that this is a possible reason for a Chihuahua to let out a loud yelp with being picked up. In addition, the shaking that you mentioned may also be due to pain.
There are many 'unseen' health concerns
that can cause quite a bit of pain; though usually there are other signs aside from yelping.
Possible reasons are:
Luxating patella -
This is not uncommon with the Chihuahua breed. Dogs can have a genetic predisposition to this and even an action as innocent as jumping down from a sofa can trigger it. With some dogs, it will be a slow, progressive condition; the hip joint and socket wear out over time.
Many dogs will act fine and only yelp out in pain when touched. If the condition is not resolved, inflammation will develop and it is usually at that point that the Chihuahua will show other signs such as limping and reluctance to exercise.
A experienced veterinarian can often diagnose this by manipulation of the leg alone; however x-rays are almost always performed to confirm the diagnosis.
Back problems -
While not probable, it is possible. There are some back problems that are acute and will go away with time. Sometimes when an injury involves the back, a Chihuahua will only yelp when the back randomly spasms or when touched, including when being picked up.
There are countless other health conditions that would show up with a combination of blood work and x-rays. If the veterinarian only performed a quick 'once-over' and did not do any testing, we would highly recommend seeking out another vet for a second opinion.