closed as too broad by ja72, Brandon Enright, Kyle Kanos, John Rennie, Qmechanic♦ Jan 6 '14 at 17:03
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I wll answer the first question to get the ball rolling.
Consider tracking a particle that is constrained on a 2D path. Can can choose to keep track of the $x$, $y$ coordinates separately (2 dof system) and include a constraint equation to keep the particle on the path.
You can use a generalized coordinate which might be the distance along the path, or the polar angle, or whatever else makes sense to you and keeps the particle on the path with only 1 coordinate specification. Thus no need for an additional constraint equation.
That is the reason people use generalized coordinates, to reduce the system degrees of freedom acording to the constraints.