I am looking for two casual nouns or adjective to describe/call somebody who is sensitive (and not sensitive) to physical pain.
From hyper-(hypo-) algesia, I suppose that one can be said to be hyper(hypo-) algesic. However, these sounds very formal. I am looking for a word that is commonly used and is eventually slightly mocking.
Do the words hyperalgesic and hypoalgesic exist?
Are hyper(hypo-) algesic formal?
Are there less formal (eventually slightly mocking) equivalent terms?
Hyperalgesic and hypoalgesic are real words, but they are only normally used in a formal, medical context. Most English speakers would not know the meaning of them. Another formal word that is related to hyperalgesic is allodynic, which means that things that should not be painful at all produce pain sensations. This blog post by a neuroscientist explains the difference quite well in the first three paragraphs.
The most common phrase pair I’ve encountered for what you are describing is “high (low) pain tolerance”. As for giving it a mocking tone, you can use quotation marks to emphasize that someone said “high”, but it may not be true:
“Apparently, he has a ‘high’ pain tolerance!”, said the nurse.
Also, there is an element of truth to the running joke in nursing circles that most people who claim to have a high pain tolerance actually have exactly the opposite.
Note that there is also another related, but separate concept to pain tolerance, pain threshold. This is the level at which a given person starts to feel pain; it is not the same thing as pain tolerance, which is the ability to cope with pain.