Making the turn at first base

Always try to make the path to the last base your running to, straight as possible. You want the shortest distance (quickest time) to the base where a play might be made.

Once a runner recognizes there will be no play at 1B he should start rounding to get the fastest path to 2B. Yes this is slower to 1B, but quicker to 2B. The fact that it is slower to 1B actually gives the runner additional time to decide on going to 2B.

It really boils down to "do you want to get from home to first quicker or do you want to get from first to second quicker?" Common sense answers most questions in baseball.

Same thing holds true at 2B trying to score. It's tougher because of the many different situations that come up, but ideally you want to create as close to a straight line as possible from 3B to home plate. Most with 2 outs will take a deeper lead off 2B (towards outfield) to get a sharper turn at 3B.

To answer your last question... To a certain extent it does depend on the runner. There is no standard width of turn that I know of other than trying to create the quickest time from 1B to 2B. There are running techniques that will help though.