I went to this place to try cyberhair. They tied three artificial hairs to one of my own hairs and did this for a total of 100 hairs. After the procedure, there was pain in the area that they worked. A few hairs came out from the procedure. Is this legitimate?

Cyberhair was developed by Aderans, a large maker of wigs and owner of hair transplant chains Bosley and Medical Hair Restoration (which has recently merged under the Bosley name). It’s just a type of synthetic hair fiber that is braided to your existing hair, a technique that has been around for many years.
The biggest problem in having artificial hair braided to your own is that in many cases it causes traction alopecia. This results in progressive hair loss from the constant tugging on the original existing hair.

A pea-sized, slightly raised bald spot where 10 to 15 hairs are sprouting? I have no clue what it could be. If your physician has retired, I’d seek out another physician for an exam. Perhaps a dermatologist would be your best bet. Based on your description though, it does not sound like male pattern balding.
If the hat is not pulling on your hair, taking it off and putting it on over and over again should not cause any impact. Same goes for hair nets.