Question 1: I'm doing very well. How about yourself?
Question 2: I'm doing very well. How about you?
Please explain the difference if both questions are correct.
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The sentence with "yourself" is not, strictly speaking, wrong. It sounds a bit folksy and colloquial, like the person who is saying this is old friends with the person he is addressing.
The thing is, these formulas that are used in greetings are set phrases. Even if an alternative is not completely wrong, it's best not to insist on using them.
A better use of "yourself" might be in an exchange like this,
– How are you?
– Fine. And yourself?
This is a shortened version of the proper phrase. Again, colloquial.
If you are asking about subtexts of meaning, using yourself underscores that you are reversing the question from you back to the person who has asked first.
How about yourself is grammatically incorrect. When using reflexive pronouns, you should use a subject at least once, although you can have an implied use to e.g.. You did that to yourself or (You) Look at yourselves. It is grammatically correct to say How about you, yourself, but that's pretty clunky.