Why does my douschund yelp when I pick her up by her upper chest or under arm area?

Maybe because it isn't the proper way to pick up a small dog, and perhaps it hurts her. You should be supporting her whole body when you pick her up. Or perhaps you should take her into the vet and have them give her a check over to be on the safe side.

A friend of mine has two long haired daschunds and I notice when they pick either one up, they support the whole body.

One of the best ways to pick up a dog is to put you hands or arms around behind and in front of their legs supporting them at both ends. You need to be very careful to never strain a Dachshund's back.

I have seen people pick them up by their collar. NO! Or by their front legs . No! Or by the middle leaving their legs to dangle. No! < not good ways to pick up a dog. I have 2 small dogs. I put my 1 hand under their chest the other hand under their stomach close to their legs and after I get them close to my body. I keep my hand under their chest while cradling their stomach on my forearm.