To feel badly implies that your sense of touch is not right. When you are referring to a sense of touch, then badly is used as an adverb describing the verb to feel or touch. The correct way to say the sentence is, "I feel bad for him because he didn't make the cut."
Correspondingly, what is correct I feel good or I feel well?
Both answers are correct. Feel well and be well refer to one's health. Well in this case is an adjective. This is the only use of well as an adjective-to mean healthy, not sick. Feel good means to feel happy.
Beside above, is it correct to say feel good?
The proper phrase is “I feel well” if you're trying to say that you're good at the act of feeling. It's a bit awkward. To me, “I feel well” means “I feel that I am well”, that my health is good. When I learned grammar, “I feel good” I learned that was grammatically improper.
How do u feel today?
"How are you feeling today" - The word feeling implies that the person is able to change from feeling to feeling say in physical wellbeing after an injury or illness. "How do you feel today" - The word feel assumes the person is stuck in a feeling - How do you feel today about what happened yesterday?
Are u feeling better now?
It is a short form of the longer question - “Are you feeling better now?” it's meant to ask if you are recovered or at least stable after whatever happened without being too intrusive.