1) In general, yes, you can use a bordered hessian to determine LOCAL maximum and minimums.
2) When seeking the GLOBAL maximum and minimum, it's easier just to identify the *possible* places where they could occur, and then just plug in values. Those possible places are stationary points (via Lagrange multipliers), places where the function isn't differentiable, and domain borders.
3. In this case, it's possible to solve the problem without doing hardly anything. You can solve it by inspection. The constraint says you're on an ellipsoid. That f has a very natural geometric meaning. Think about it for a minute, and you can just say - no work at all - what the global maximum and minimum are, and all the points where they occur.