1) I've used clicker training to tame/bond with all of my birds. I find it extremely effective BUT it's not just a matter of buying a clicker. Clicker training is only effective if you are willing to learn about the principles of positive reinforcement training, and how/why a clicker is used. A clicker is just a tool, and so in the hands of the ignorant will be useless. Positive reinforcement training on the other hand is possible without a clicker, so once you've learned how to use this technique it's up to you to decide whether or not a clicker is right for you. Without going into all of the details I can say that a clicker (or "bridge") is going to be most useful for teaching finely-nuanced behaviors in which timing and "capturing" the initial behavior may be difficult. I've always just started out with it however to get my birds used to the clicker. For a crash course in positive reinforcement training using a clicker definitely pick up the book "Getting Started: Clicker Training for Birds" by Melinda Johnson.
2) The "best" method IMO is the one that does not force your bird to interact with you against its will. Positive reinforcement is one of these, but it also requires a lot of patience and understanding.
3) Train frequently and in short bursts (so 2-3x a day for 5-15 mins at a time). Make an effort to learn your bird's body language early and try to respect what they tell you! Some birds will need weeks to settle in, or have "good" days and "bad" days. The more you push when it's obvious they are cranky or uncomfortable the more you will encourage biting or other fear responses. In general try to make all of your interactions positive! Don't sit there with your hand in the cage if it makes your bird too scared to move around and eat. Instead give them space and treats! Let them tell you what pace they are comfortable with. Finally, every bird is an individual so throw out all of your expectations and preconceived timelines. No, your bird does not have to be tame in 3 days, or 3 weeks, or even 3 months! If you worry too much about achieving certain goals by some arbitrary time, you're only going to get frustrated and set yourself up for failure.
4) See above. None of our budgies have ever been hand-fed. The last was neglected for almost 2 years and came to us extremely phobic and distrustful of hands. The ONLY thing that caused him to turn around was having patience. He did not willingly sit on my finger for almost 6 months. So the process can be slow, but very rewarding in the end.