That would be a way to ask your friends how they are doing, but with the doctor, your teacher, a political figure, you would use the formal you which requires third person singular conjugation.
"The formal you?" Where is that in the conjugation of verbs? I understand what you are talking about because I have studied Spanish, but this appears nowhere in Duolingo in the Italian section that I have seen!
is there a difference between io sono and io sto (if this is the right form of this word)?
I can't say this for sure, but knowing what I know of Spanish I'm assuming io sono=I am in the sense of something definite. Example: I am Samantha=Sono Samantha. For io sto=something indefinite like emotion/how ones feeling at that particular moment, etc. Ex: I am well= sto bene.
Usually "essere" (io sono) is used for identity, qualities, permanent attributes, etc. while "stare" (io sto) is for locations, in the continuous tense and in idiomatic expressions like "how do you do" (come stai) but don't take it as a rule - there are a lot of other uses and exceptions.
For variety, I used "How do you do" as an equivalent to "how are you". I was NOT accepted
"How do you do" is a very strange phenomenon in English. You use it only when you meet someone for the first time - ever. (After that you use: "How are you?" etc.) But this is all very, very formal. Just don't use it to ask how someone is. Trivia of languages are so strange, no?
I believe "come sta" is used in formal situations where third person is used. "Come stai" directly means "how are you" and is more informal and common.