You would NEVER ask someone you just met 'tell me about yourself'. That question is way too open and broad so you are correct with your assessment. That question is sometimes used in job interviews but then employers are purposely asking tough questions to see how prospective employees react. I don't think I have EVER asked that question. If you want to get to know someone or just make small talk asking specific questions would be preferable. "How long have you lived here", "Where did you go to school" etc. Asking a person to tell you about themselves will show your inability to make conversation and definitely make them a bit uncomfortable.

Why don't we sit down and relax? Then have a talk

You're right, it is customary to give a piece of information before you ask for a piece of information. If I were B I would give only basic information like my name, and what country I am in, for example: B: My name's Lee, I live in the UK. What about you?