Yes, this is a bit confusing, because English speakers are a little unclear.
1. Depends whether you want the year included. Strictly speaking, a date of birth includes the year, but you might not actually *need* to know their age, so asking for their *full* date of birth might be more information than you need and more information than they're comfortable giving. So it depends. If you *do* want the year then "What is your date of birth?" is the question, and they'll tell you the date including the year (if they're comfortable doing so). If you don't need the year, then just ask "when's your birthday" and they'll just give you the month and day-of-the-month.
2. No. But this is where English speakers get vague. I say "no" because when native English speakers say "birthday" they do not mean "the year/month/day on which you were born", they mean "any anniversary of the date of your birth." If you are filling in an official form, then you are NOT asked for your "birthday"; you are asked for your "date of birth". A date of birth is year/month/day. A birthday is the anniversary of your birth, so by definition it does not include a year. When a friend wants to know when they need to give you a gift, they will not ask for your "date of birth"; they will ask "when is your birthday?", in other words "what date each year marks the anniversary of the day you were born." They only want month/day. So, you, see why I answered "no" to this question. "Does the date of birth mean the birthday?". No, it does not. :)
3. I'm unsure what kind of related questions you mean. If you mean the questions we've been talking about, then those are not "related" questions, those are "those questions" or "And how would I answer these same questions?" Hopefully I've shown that.