In the U.S., this is a polite and respectful way of greeting someone. You are not being nosy or rude, but you are expressing concern for their well-being, as "how are you?" means "how are you feeling?"
Question: "Hi, how are you?"
Answer: "Hi, I'm good, thank you." (or "I'm fine, thanks.")
You can also add after your reply: "And how are you?" This is very polite, but it is not required.
You do not need to tell people how you are feeling or share anything personal. It is just a way of expressing politeness and respect when you greet someone.
And if you are not feeling great, it is bad manners in most situations to reply honestly and say, "I'm awful," or "I'm not so good," although you CAN be this candid with friends and family. For example:
Question: "Hi Mom, how are you?"
Answer: "I feel terrible! I couldn't sleep at all last night."
Normally, people just automatically say, "I'm fine, thanks," even if they are having a bad day. Again, this is just part of our cultural traditions for polite greetings.
"How are you?" and "How are you doing?" mean the same thing - they are used interchangeably.
I hope this helped!