Your destiny number (bhagyank), summed from your full date of birth, points to your life's broad direction, lessons and purpose.
Every digit of your full birth date is added and reduced to a single digit, so day, month and year all contribute. Because it uses the whole date it is more specific than the root number, and it shifts the emphasis from how you start things to the direction you keep returning to across a life.
The destiny number carries a ruling graha — 1 Sun through 9 Mars — and that is the useful handle. If your destiny number points at Jupiter, the question worth asking is where Jupiter actually sits in your chart and how strong it is, because a number pointing at a debilitated planet reads very differently from one pointing at an exalted one. The free kundli answers that in a minute.
Yes — they are two names for the same calculation, the full birth date reduced to a single digit. Western sources tend to say life path, Indian sources bhagyank.
No. It comes from your birth date, which does not change. Name numbers change with spelling; date-based numbers do not change at all.