1. How are you doing?

2. How are you feeling?
3. How is it going?
4. How are you?
5. How do you feel?
6. How are you today?

- I feel delighted.
- I feel happy.
- I feel satisfied.
--------------------------------
Are the questions all the same? Can we answer as in the above?

2 answers

Yes, they all mean much the same thing, and they can be answered with any of the three possible responses. One might also say, "I feel terrible!" :)
Yes, the questions are all fairly standard ways to greet other people.

The replies are usually one or two words --
Oh, fine
Okay
Just fine
Doing well
(And others)