How do I wrap a road bike handlebar?
Park Tools has a great description with a video, pictures, and instructions.

Expanding the Park link a little:

1 - you can start at either end, conventionally start at the bar ends because that way the overlap on the wraps has the lower side out so smooths as you slide your hands down the bar. That way it lasts longer and looks nicer.

2 - wrap up the outside on both sides. So the answer to your question is "yes".

3 - this is where most of the variation in styling occurs. Tape that is more flexible or stretchy can have fewer wraps around the brake levers, although you can often do a single skip around the back of the lever even with quite unyeilding tape. I prefer to go once around the levers to get 3 strips of tape behind them. Experiment to see what works for you.

If you're wrapping bars for the first time I suggest buying the cheapest tape you can find to experiment with, as well as the stuff you actually want. It's something that's not too hard but you will probably make mistakes the first time.